id,status,shape,domain,statement,sourced_how,cheapest_test,source_url,claimed_by
ax-2604.09805-0b3b2ee8,open,needs-theory,computer science / cs.SE,5 Open Questions: What mechanisms best support the transition from human-supervised to autonomous agent operation?,enumerated in arXiv paper titled 'open problems' (prose (list items and question sentences under an open-problems heading)): Building an Internal Coding Agent at Zup: Lessons and Open Questions,,https://arxiv.org/abs/2604.09805,
ax-2604.09805-20f4b467,open,needs-theory,computer science / cs.SE,5 Open Questions: How can safety policies be specified and enforced consistently across tools with overlapping capabilities?,enumerated in arXiv paper titled 'open problems' (prose (list items and question sentences under an open-problems heading)): Building an Internal Coding Agent at Zup: Lessons and Open Questions,,https://arxiv.org/abs/2604.09805,
ax-2604.09805-2a59c522,open,needs-theory,computer science / cs.SE,"5 Open Questions: What principles, metrics, or evaluation frameworks can guide the systematic design of tool interfaces for LLM-based agents?",enumerated in arXiv paper titled 'open problems' (prose (list items and question sentences under an open-problems heading)): Building an Internal Coding Agent at Zup: Lessons and Open Questions,,https://arxiv.org/abs/2604.09805,
ax-2604.09805-2bfdc523,open,needs-theory,computer science / cs.SE,5 Open Questions: How should tool manifests be designed to minimize model misuse and maximize correct invocation?,enumerated in arXiv paper titled 'open problems' (prose (list items and question sentences under an open-problems heading)): Building an Internal Coding Agent at Zup: Lessons and Open Questions,,https://arxiv.org/abs/2604.09805,
ax-2604.09805-527aa52d,open,needs-theory,computer science / cs.SE,"5 Open Questions: What memory architectures balance persistence, retrieval accuracy, and staleness management across sessions—and how should agents decide what to retain, forget, or update over time without explicit user instruction?",enumerated in arXiv paper titled 'open problems' (prose (list items and question sentences under an open-problems heading)): Building an Internal Coding Agent at Zup: Lessons and Open Questions,,https://arxiv.org/abs/2604.09805,
ax-2604.09805-5f6ec5ec,open,needs-theory,computer science / cs.SE,5 Open Questions: How should agent-generated code be integrated into existing quality assurance pipelines?,enumerated in arXiv paper titled 'open problems' (prose (list items and question sentences under an open-problems heading)): Building an Internal Coding Agent at Zup: Lessons and Open Questions,,https://arxiv.org/abs/2604.09805,
ax-2604.09805-73d3e13c,open,needs-theory,computer science / cs.SE,5 Open Questions: How should agent memory be structured to support long-term learning without compromising session reliability?,enumerated in arXiv paper titled 'open problems' (prose (list items and question sentences under an open-problems heading)): Building an Internal Coding Agent at Zup: Lessons and Open Questions,,https://arxiv.org/abs/2604.09805,
ax-2604.09805-a1b14cfd,open,needs-theory,computer science / cs.SE,"5 Open Questions: What formalisms or policy languages can express cross-tool safety constraints in a way that is complete, verifiable, and maintainable as the tool manifest grows?",enumerated in arXiv paper titled 'open problems' (prose (list items and question sentences under an open-problems heading)): Building an Internal Coding Agent at Zup: Lessons and Open Questions,,https://arxiv.org/abs/2604.09805,
ax-2604.09805-acb9e76b,open,needs-theory,computer science / cs.SE,"5 Open Questions: Under what conditions should reasoning responsibility shift from the model to the orchestrator (or vice versa), and how does this boundary affect reliability, latency, and safety as model capabilities evolve?",enumerated in arXiv paper titled 'open problems' (prose (list items and question sentences under an open-problems heading)): Building an Internal Coding Agent at Zup: Lessons and Open Questions,,https://arxiv.org/abs/2604.09805,
ax-2604.09805-ca98d52a,open,compute-checkable-small-cases,computer science / cs.SE,"5 Open Questions: Are there adaptive trust models—based on task complexity, historical success rate, or action reversibility —that can dynamically calibrate the level of human oversight required, rather than relying on a binary mode switch?",enumerated in arXiv paper titled 'open problems' (prose (list items and question sentences under an open-problems heading)): Building an Internal Coding Agent at Zup: Lessons and Open Questions,,https://arxiv.org/abs/2604.09805,
ax-2604.09805-f11489d4,open,needs-theory,computer science / cs.SE,5 Open Questions: What is the optimal boundary between model-delegated reasoning and orchestrator-enforced control?,enumerated in arXiv paper titled 'open problems' (prose (list items and question sentences under an open-problems heading)): Building an Internal Coding Agent at Zup: Lessons and Open Questions,,https://arxiv.org/abs/2604.09805,
ax-2604.09805-fee20453,open,needs-theory,computer science / cs.SE,"5 Open Questions: What additional verification mechanisms are needed, and how should they differ from those applied to human-authored code?",enumerated in arXiv paper titled 'open problems' (prose (list items and question sentences under an open-problems heading)): Building an Internal Coding Agent at Zup: Lessons and Open Questions,,https://arxiv.org/abs/2604.09805,
ax-2605.05076-0bcb7602,open,needs-theory,mathematics / math.ST,"2.5 Further open directions: • Sample size, computation, and privacy. Understanding whether additional samples can compensate for computational constraints, and how privacy requirements interact with both statistical and computational limitations, remains an emerging and largely open area.",enumerated in arXiv paper titled 'open problems' (prose (list items and question sentences under an open-problems heading)): High-Dimensional Statistics: Reflections on Progress and Open Problems,,https://arxiv.org/abs/2605.05076,
ax-2605.05076-b6a600e3,open,needs-theory,mathematics / math.ST,"2.5 Further open directions: • Unifying frameworks. Understanding when different frameworks (statistical queries, low-degree polynomials, sum-of-squares, reductions) agree or disagree in their predictions, and developing unified theories that capture the strengths of multiple approaches, remains an important open problem.",enumerated in arXiv paper titled 'open problems' (prose (list items and question sentences under an open-problems heading)): High-Dimensional Statistics: Reflections on Progress and Open Problems,,https://arxiv.org/abs/2605.05076,
ax-2605.05076-e89f0364,open,needs-theory,mathematics / math.ST,"2.5 Further open directions: • Gaps at the level of constants. Most existing work characterizes computational-statistical gaps in terms of rates. However, for problems such as estimating a rank-1 spike in a Gaussian matrix, the gap between information-theoretically optimal and computationally efficient methods manifests at the level of sharp constants—specifically, the precise SNR threshold. It is widely believed that approximate message passing (AMP) algorithms achieve the optimal SNR threshold among efficient methods, but rigorous evidence remains limited: current low-degree lower bounds only rule out constant-degree polynomials rather than the logarithmic-degree polynomials that would provide stronger evidence. Understanding when AMP is optimal among efficient procedures, and extending constant-level gap analyses more broadly, is an important open direction. See for recent progress in this direction.",enumerated in arXiv paper titled 'open problems' (prose (list items and question sentences under an open-problems heading)): High-Dimensional Statistics: Reflections on Progress and Open Problems,,https://arxiv.org/abs/2605.05076,
ax-2605.29885-955188af,open,compute-checkable-small-cases,computer science / cs.LG,"Open Problem 1 (Formal Separation of Geometric and Algorithmic Compression): The core theoretical challenge is to establish a strict separation theorem between geometric capacity control (e.g., norm-based regularization, Rademacher complexity, margin bounds) and algorithmic compression. Can we formally characterize the hypothesis classes—such as discrete algebraic rules or formal languages—where all purely geometric generalization bounds are provably vacuous (requiring exponential sample complexity), yet differentiable algorithmic complexity measures yield tight, polynomial guarantees? Mapping the exact territories where geometric surrogates fundamentally fail, and where only algorithmic compression succeeds, remains a critical missing foundation in SLT.",enumerated in arXiv paper titled 'open problems' (formal problem environment): Open Problem: Separating Geometric and Algorithmic Compression via Cayley-Table Completion,,https://arxiv.org/abs/2605.29885,
ax-2605.29885-9bb5e691,open,compute-checkable-small-cases,computer science / cs.LG,"Open Problem 2 (Sample Complexity of the Canonical Testbed): Cayley-table completion serves as the immediate, concrete testbed for establishing this separation. Because finite groups are linearly full-rank ( $r=n$ ), applying standard matrix completion theory yields vacuous bounds exceeding $\mathcal{O}(n^{2})$ . Can we formally establish an exact recovery guarantee showing that the global minimizer of the flatness-regularized empirical loss perfectly completes the group table with high probability given only $\mathcal{O}(n\log n)$ uniformly sampled entries? Solving this would provide the first rigorous mathematical proof of the separation demanded in Open Problem 1.",enumerated in arXiv paper titled 'open problems' (formal problem environment): Open Problem: Separating Geometric and Algorithmic Compression via Cayley-Table Completion,,https://arxiv.org/abs/2605.29885,
ax-2605.30389-1e86dbdc,open,compute-checkable-small-cases,computer science / cs.FL,"Question 2: Is it true that, for every alphabet $\Sigma$ with at least two symbols, $\operatorname{ID}_{\Sigma}(p)=2|p|-\#\operatorname{var}(p)-1,$ (1) where $|p|$ is the length, namely the number of variables and constants in $p$ , and $\#\operatorname{var}(p)$ is the number of distinct variables in $p$ ?",enumerated in arXiv paper titled 'open problems' (formal problem environment): The Inclusion Depth of Pattern Languages: An Open Problem in Algorithmic Learning Theory,,https://arxiv.org/abs/2605.30389,
ax-2605.30389-3742c7a5,open,compute-checkable-small-cases,computer science / cs.FL,"Question 1: Is there an algorithm to compute $\operatorname{ID}_{\Sigma}(p)$ for any pattern $p$ and any alphabet $\Sigma$ ? If yes, is there a polynomial-time algorithm?",enumerated in arXiv paper titled 'open problems' (formal problem environment): The Inclusion Depth of Pattern Languages: An Open Problem in Algorithmic Learning Theory,,https://arxiv.org/abs/2605.30389,
ax-2605.30389-b8d5e45f,open,needs-theory,computer science / cs.FL,"Conjecture 1: Let $p$ and $q$ be two patterns. If $L(p)\subset L(q)$ , then $2|p|-\#\operatorname{var}(p)>2|q|-\#\operatorname{var}(q).$ (2)",enumerated in arXiv paper titled 'open problems' (formal problem environment): The Inclusion Depth of Pattern Languages: An Open Problem in Algorithmic Learning Theory,,https://arxiv.org/abs/2605.30389,
ax-2606.07327-188c3d0a,open,unclassified,condensed matter physics / cond-mat.mtrl-sci,What new physics can be and are already being discovered? (The issues listed above can be potentially addressed by recent algorithmic),enumerated in arXiv paper titled 'open problems' (section headings phrased as questions): Six Open Questions in Machine-Learned Interatomic Potential Foundation Models,,https://arxiv.org/abs/2606.07327,
ax-2606.07327-2d7f8aee,open,unclassified,condensed matter physics / cond-mat.mtrl-sci,"What is the minimal definition of an atomistic foundation model? (Classical force fields are powerful but inherently specialized, generally requiring substantial domain expertise, hand-crafted functional forms, and parameterization to tailor them to specific chemistries or thermodynamic conditions. This bespoke nature makes them effective for targeted tasks but fundamentally unscalable. Foundation MLIPs, in contrast, aspire to transcend this fragmentation. The…)",enumerated in arXiv paper titled 'open problems' (section headings phrased as questions): Six Open Questions in Machine-Learned Interatomic Potential Foundation Models,,https://arxiv.org/abs/2606.07327,
ax-2606.07327-40367028,open,unclassified,condensed matter physics / cond-mat.mtrl-sci,"Do we need more data, better data, or better models? (A central tension in the development of MLIPs echoes a long-standing debate in AI: should progress come primarily from carefully designed inductive biases or from scaling models and data? Sutton’s “Bitter Lesson” argues that, in the long run, systems that rely less on human-engineered structure and more on compute-enabled general methods tend to outperform those with strong built-in priors. In…)",enumerated in arXiv paper titled 'open problems' (section headings phrased as questions): Six Open Questions in Machine-Learned Interatomic Potential Foundation Models,,https://arxiv.org/abs/2606.07327,
ax-2606.07327-558bc342,open,unclassified,condensed matter physics / cond-mat.mtrl-sci,"Why do models fail to capture long-range interactions? (Message Passing Neural Networks (MPNNs), one of the most widely used architectures for MLIPs, exhibit a local inductive bias , as message aggregation for each node is typically limited to its neighbourhood. This locality reflects the assumption that short-range interactions (up to a large enough receptive field) govern the physics, while taking into account the limitations due to computing costs:…)",enumerated in arXiv paper titled 'open problems' (section headings phrased as questions): Six Open Questions in Machine-Learned Interatomic Potential Foundation Models,,https://arxiv.org/abs/2606.07327,
ax-2606.07327-9b80a7a3,open,unclassified,condensed matter physics / cond-mat.mtrl-sci,"Can MLIPs discover truly new physics? (A defining aspiration of atomistic foundation models is that they should not merely interpolate within familiar chemistry but also generalise to new regimes, uncovering behaviours or mechanisms that are absent from their training data. Such capability can arise via two)",enumerated in arXiv paper titled 'open problems' (section headings phrased as questions): Six Open Questions in Machine-Learned Interatomic Potential Foundation Models,,https://arxiv.org/abs/2606.07327,
ax-2606.07327-b8f89180,open,unclassified,condensed matter physics / cond-mat.mtrl-sci,"What are long-range interactions? (From the above discussion, it is clear that the inclusion of long-range interactions is needed to describe the physics of many systems. There are many different ways to define long-range interactions, which broadly fall into two categories: those based on physics and those based on the graph/architecture of the)",enumerated in arXiv paper titled 'open problems' (section headings phrased as questions): Six Open Questions in Machine-Learned Interatomic Potential Foundation Models,,https://arxiv.org/abs/2606.07327,
ax-2606.07327-c52467f0,open,unclassified,condensed matter physics / cond-mat.mtrl-sci,"What needs to happen for atomistic foundational models to discover new physics? (MLIP frameworks today learn well within known chemistry and physics, but struggle outside training distributions, largely because they provide approximate representations of the underlying bonding physics in molecular systems by interpolating within the range of the training data . This means that MLIP approaches are often poorly equipped to capture rare events or phenomena that occur under…)",enumerated in arXiv paper titled 'open problems' (section headings phrased as questions): Six Open Questions in Machine-Learned Interatomic Potential Foundation Models,,https://arxiv.org/abs/2606.07327,
ax-2606.07327-e19bc0ac,open,unclassified,condensed matter physics / cond-mat.mtrl-sci,"Can MLIPs scale to do more useful simulations? (In the context of MLIPs, scalability may refer to the ability to handle larger atomistic systems, longer simulation timescales, larger and more expressive models, or higher-throughput ensemble simulations. Current foundation models, such as MACE-MP , have shown success in many applications, ranging from solid-state electrolytes and heterogeneous catalysis to organic drug-like molecules . However,…)",enumerated in arXiv paper titled 'open problems' (section headings phrased as questions): Six Open Questions in Machine-Learned Interatomic Potential Foundation Models,,https://arxiv.org/abs/2606.07327,
ax-2606.07327-ed49aa87,open,unclassified,condensed matter physics / cond-mat.mtrl-sci,"Can MLIPs really handle long-range interactions, and does it matter? (As discussed previously, one approach to generating better models is to include long-range interactions. The locality of interactions in MLIPs is in part justified by Kohn’s ‘near-sightedness’ principle . However, there are many different systems where it has been shown in the literature that it is beneficial to include long-range interactions to fully describe the physics of the system: notably,…)",enumerated in arXiv paper titled 'open problems' (section headings phrased as questions): Six Open Questions in Machine-Learned Interatomic Potential Foundation Models,,https://arxiv.org/abs/2606.07327,
ax-2607.04129-02d02142,open,needs-theory,computer science / cs.SE,V Open Problems: For Practitioners: (PP2) Enforcing the implementation–model contract: How can the implementation be shaped so that it faithfully realizes the theoretical task model?,enumerated in arXiv paper titled 'open problems' (prose (list items and question sentences under an open-problems heading)): Toward the Right Analytical Model and System Software for Autonomous Driving Systems: Open Problems and Research Directi,,https://arxiv.org/abs/2607.04129,
ax-2607.04129-1836a5e0,open,compute-checkable-small-cases,computer science / cs.SE,V Open Problems: For Practitioners: (PP1) Eliminating non-essential complexity: How can we remove the non-essential complexity that the middleware layer imposes on scheduling?,enumerated in arXiv paper titled 'open problems' (prose (list items and question sentences under an open-problems heading)): Toward the Right Analytical Model and System Software for Autonomous Driving Systems: Open Problems and Research Directi,,https://arxiv.org/abs/2607.04129,
ax-2607.04129-378f9b24,open,needs-theory,computer science / cs.SE,V Open Problems: For Practitioners: (PP5) Implementing state-of-the-art schedulers: How can the latest scheduling algorithms actually be enforced on commodity middleware and OS?,enumerated in arXiv paper titled 'open problems' (prose (list items and question sentences under an open-problems heading)): Toward the Right Analytical Model and System Software for Autonomous Driving Systems: Open Problems and Research Directi,,https://arxiv.org/abs/2607.04129,
ax-2607.04129-56960ff9,open,needs-theory,computer science / cs.SE,"IV Open Problems: For Theorists: (TP3) Heterogeneous resource models: How can a timing model explicitly capture GPU/NPU execution, memory bandwidth, and on-chip interconnect, rather than leaving them unmodeled or abstracted away?",enumerated in arXiv paper titled 'open problems' (prose (list items and question sentences under an open-problems heading)): Toward the Right Analytical Model and System Software for Autonomous Driving Systems: Open Problems and Research Directi,,https://arxiv.org/abs/2607.04129,
ax-2607.04129-7e0a155f,open,needs-theory,computer science / cs.SE,"V Open Problems: For Practitioners: (PP4) Detecting timing-constraint violations early: How can we detect, at runtime and early enough to act, that a timing constraint will be missed?",enumerated in arXiv paper titled 'open problems' (prose (list items and question sentences under an open-problems heading)): Toward the Right Analytical Model and System Software for Autonomous Driving Systems: Open Problems and Research Directi,,https://arxiv.org/abs/2607.04129,
ax-2607.04129-87712eb6,open,needs-theory,computer science / cs.SE,"IV Open Problems: For Theorists: (TP5) Binding schedulability to safety across the MRM mode change: What does a timing violation mean for safety, and what must a timing analysis guarantee when a miss becomes imminent?",enumerated in arXiv paper titled 'open problems' (prose (list items and question sentences under an open-problems heading)): Toward the Right Analytical Model and System Software for Autonomous Driving Systems: Open Problems and Research Directi,,https://arxiv.org/abs/2607.04129,
ax-2607.04129-9b1a2d28,open,needs-theory,computer science / cs.SE,"V Open Problems: For Practitioners: (PP3) Separating functions of different criticality: How can functions of different criticality (the safety-critical MRM at the highest level, the nominal AD functions of perception, planning, and control at an intermediate level, and logging or a human-machine interface at the lowest) share one platform without the lower-criticality ones disturbing the higher-criticality ones?",enumerated in arXiv paper titled 'open problems' (prose (list items and question sentences under an open-problems heading)): Toward the Right Analytical Model and System Software for Autonomous Driving Systems: Open Problems and Research Directi,,https://arxiv.org/abs/2607.04129,
ax-2607.04129-bef3de69,open,needs-theory,computer science / cs.SE,"IV Open Problems: For Theorists: (TP1) The right unit of analysis: What should be the unit of analysis: callback, node, chain, E2E cause-effect chain, or graph?",enumerated in arXiv paper titled 'open problems' (prose (list items and question sentences under an open-problems heading)): Toward the Right Analytical Model and System Software for Autonomous Driving Systems: Open Problems and Research Directi,,https://arxiv.org/abs/2607.04129,
ax-2607.04129-d62a8989,open,needs-theory,computer science / cs.SE,"IV Open Problems: For Theorists: (TP4) Execution-time models under scene-/data-dependent and hardware-induced variability: How should execution time be modeled when it depends on scene, data volume, and deep neural network (DNN) pre-/post-processing?",enumerated in arXiv paper titled 'open problems' (prose (list items and question sentences under an open-problems heading)): Toward the Right Analytical Model and System Software for Autonomous Driving Systems: Open Problems and Research Directi,,https://arxiv.org/abs/2607.04129,
ax-2607.04129-e8868263,open,needs-theory,computer science / cs.SE,"IV Open Problems: For Theorists: (TP2) Joint analysis of heterogeneous timing metrics: How can one analysis reason jointly about E2E latency, response time, freshness, timing disparity, and miss probability?",enumerated in arXiv paper titled 'open problems' (prose (list items and question sentences under an open-problems heading)): Toward the Right Analytical Model and System Software for Autonomous Driving Systems: Open Problems and Research Directi,,https://arxiv.org/abs/2607.04129,
ax-2607.07851-105112d7,open,unclassified,mathematical physics / math-ph,"Problem 4.11 (Volume-preserving vs. symplectic for $n\geq 2$ ): Corollary 4.4 pins down the measure but, for $n\geq 2$ , not the finer symplectic structure: $\mathrm{SDiff}(\mathbb{R}^{2n})\supsetneq\mathrm{Symp}(\mathbb{R}^{2n})$ . Identify the weakest statistical requirement that reduces the invariance group from volume-preserving to symplectic. Candidates, in increasing strength: (i) invariance of all within-DOF uncertainties $u_{j}$ at equilibrium; (ii) invariance of the linear symplectic capacity of covariance ellipsoids (see the capacity remark 3.4 ; by this fails for generic volume-preserving linear maps once $n\geq 2$ ); (iii) invariance of the whole symplectic spectrum (Problem 4.8 ). Determine which of (i)–(iii) are equivalent, and which are estimable from kime-tomographic data in the sense of Problem 3.18 .","enumerated in arXiv paper titled 'open problems' (formal problem environment): Kime-Representation Formulations of Three Open Problems in the Foundations of Classical Mechanics: Uncertainty, Invarian",,https://arxiv.org/abs/2607.07851,
ax-2607.07851-15e7560c,open,unclassified,mathematical physics / math-ph,"Problem 3.17 (Symplectic Schur–Horn problem for within-DOF uncertainties): Characterize, for fixed symplectic spectrum $\nu_{1}\geq\cdots\geq\nu_{n}>0$ , the attainable set $\mathcal{U}(\nu)=\Bigl\{\bigl(u_{1}(\Sigma),\dots,u_{n}(\Sigma)\bigr):\Sigma\in\text{the }\operatorname{Sp}(2n,\mathbb{R})\text{-orbit with spectrum }\nu\Bigr\}\subset\mathbb{R}_{>0}^{n},$ where $u_{j}(\Sigma)=\sqrt{\det\Sigma_{jj}}$ . In particular, we need to explore (a) is $\min_{j}u_{j}\geq\nu_{n}$ on the whole orbit (so that no DOF can be squeezed below the smallest symplectic eigenvalue)? (b) in the equipartitioned case $\nu_{j}\equiv\nu$ , is $u_{j}\geq\nu$ for every $j$ (the per-DOF form of the conjecture in )? And (c) describe $\mathcal{U}(\nu)$ by majorization-type inequalities, in analogy with the Schur–Horn theorem, using the symplectic eigenvalue technology of . Some partial cues include (i) $\prod_{j}u_{j}\geq\prod_{j}\nu_{j}$ (Theorem 3.16 ); (ii) $u_{j}\geq\nu_{n}$ would follow from the interlacing-type bound “every $2\times 2$ symplectic compression of $\Sigma$ has symplectic eigenvalue $\geq\nu_{n}$ ,” a statement of exactly the kind studied in ; and (iii) for $n=1$ , $\mathcal{U}(\nu)=\{\nu\}$ (Theorem 3.16 ).","enumerated in arXiv paper titled 'open problems' (formal problem environment): Kime-Representation Formulations of Three Open Problems in the Foundations of Classical Mechanics: Uncertainty, Invarian",,https://arxiv.org/abs/2607.07851,
ax-2607.07851-22b95231,open,unclassified,mathematical physics / math-ph,"Problem 3.19 (Nonlinear invariant interpolating entropy and capacity): Construct a functional $\mathcal{C}[\rho]$ of states on $\mathbb{R}^{2n}$ such that: (i) $\mathcal{C}$ is invariant under all (possibly nonlinear) Hamiltonian flows; (ii) $\mathcal{C}$ reduces to $\pi\nu_{n}$ on Gaussian states; (iii) $\mathcal{C}$ lower-bounds $2\pi\min_{j}u_{j}$ up to a universal constant; and (iv) $\mathcal{C}$ is expressible through the kime-torus data of Problem 3.18 (hence estimable). Consider the following candidate functional, a sublevel-set capacity of $\rho$ at the entropy-calibrated level $e^{-\mathsf{S}[\rho]}$ , i.e., $\mathcal{C}[\rho]=c\bigl(\{\rho\geq e^{-\mathsf{S}[\rho]}\}\bigr)$ for a normalized symplectic capacity $c$ . Properties (i) and (ii) then hold by symplectomorphism-invariance of $c$ and a direct Gaussian computation, while (iii)–(iv) are open.","enumerated in arXiv paper titled 'open problems' (formal problem environment): Kime-Representation Formulations of Three Open Problems in the Foundations of Classical Mechanics: Uncertainty, Invarian",,https://arxiv.org/abs/2607.07851,
ax-2607.07851-51bcfa22,open,unclassified,mathematical physics / math-ph,"Problem 4.9 (Entropy-only rigidity): Theorem 4.1 assumed entropy invariance for all states. Determine the minimal state classes $\mathcal{F}$ for which “ $\mathsf{S}[f_{*}\rho]=\mathsf{S}[\rho]$ for all $\rho\in\mathcal{F}$ ” still forces $J_{f}\equiv 1$ (e.g., Gaussians only; kime states with von Mises phase laws only), and, dually, characterize the group of transformations preserving the entropy of every equilibrium (phase-equipartitioned) kime state. The latter group is strictly larger than the measure-preserving group (it contains all fiberwise rotations $\theta\mapsto\theta+c(J)$ trivially, but also non-measure-preserving maps acting only on null sets of equilibria); its computation quantifies exactly how much of the symplectic structure is visible to equilibrium thermodynamics alone.","enumerated in arXiv paper titled 'open problems' (formal problem environment): Kime-Representation Formulations of Three Open Problems in the Foundations of Classical Mechanics: Uncertainty, Invarian",,https://arxiv.org/abs/2607.07851,
ax-2607.07851-7e63782b,open,unclassified,mathematical physics / math-ph,"Problem 4.10 (Generalized complex rigidity of the Wick interpolation): Proposition 4.5 produces a Kähler triple in one kime DOF; Proposition 4.6 deforms the dynamics between its symplectic and metric legs. Formulate and prove (or refute) the following rigidity statement: if a $2n$ -dimensional state continuum carries (a) a reparametrization-invariant entropy (hence, by Theorems 4.1 and 4.3 , a distinguished volume), (b) a one-parameter interpolation of evolutions that is entropy-conserving at one end and satisfies a de Bruijn identity $\frac{\mathrm{d}}{\mathrm{d}t}\mathsf{S}=\varepsilon\,\mathcal{I}\geq 0$ elsewhere (Theorem 3.21 (ii)), then the infinitesimal generators assemble into a generalized complex (indeed generalized Kähler) structure in the sense of , whose pure-symplectic locus is the Hamiltonian sector and whose type jumps encode the diffusive sector. A positive answer would derive “quantities pair inside $\mathbb{C}$ ” from entropy axioms alone, completing the program of Problem (II).","enumerated in arXiv paper titled 'open problems' (formal problem environment): Kime-Representation Formulations of Three Open Problems in the Foundations of Classical Mechanics: Uncertainty, Invarian",,https://arxiv.org/abs/2607.07851,
ax-2607.07851-7fc5cf8f,open,unclassified,mathematical physics / math-ph,"Problem 5.10 (Sharp uncertainty relation on $\mathcal{O}_{m,s}$ ): Prove the relativistic extension of Theorem 5.3 : an entropic inequality on $\mathcal{O}_{m,s}$ , invariant under the Poincaré group and under all Hamiltonian flows, that reduces to ( 13 ) on the little-group fiber in the rest frame and to ( 4 ) on the translational factor in the nonrelativistic limit. Identify the extremal family (conjecturally: relativistic Gaussian in $(x,p)$ $\otimes$ von Mises–Fisher in the direction, coupled only through the Tulczyjew constraint) and the role of the two Casimirs as the invariant scales, with $W\cdot W=-m^{2}c^{2}s^{2}$ playing the part of $\prod_{j}\nu_{j}$ in Theorem 3.16 .","enumerated in arXiv paper titled 'open problems' (formal problem environment): Kime-Representation Formulations of Three Open Problems in the Foundations of Classical Mechanics: Uncertainty, Invarian",,https://arxiv.org/abs/2607.07851,
ax-2607.07851-8ad272ac,open,unclassified,mathematical physics / math-ph,"Conjecture 3.22 (Equipartition duality): Fix $n\geq 2$ , an entropy value $s$ , and an action marginal $\rho_{\bm{J}}$ on $(0,\infty)^{n}$ . Among all states on $\mathbb{T}^{n}\times(0,\infty)^{n}$ with entropy $\geq s$ and action marginal $\rho_{\bm{J}}$ , the phase-equipartitioned product state (unique when it exists) simultaneously (i) maximizes the entropy, (ii) minimizes every within-DOF uncertainty $u_{j}$ , and (iii) is the unique state at which the per-DOF conjectured bound of Problem 3.17 (b) is saturated for all $j$ ; moreover it is the unique fixed point, with the given marginal, of the multi-DOF kime-deformed semigroup $\partial_{t}\tilde{\rho}=\sum_{j}(-\omega_{j}\partial_{\theta_{j}}+\varepsilon\,\partial^{2}_{\theta_{j}})\tilde{\rho}$ .","enumerated in arXiv paper titled 'open problems' (formal problem environment): Kime-Representation Formulations of Three Open Problems in the Foundations of Classical Mechanics: Uncertainty, Invarian",,https://arxiv.org/abs/2607.07851,
ax-2607.07851-b83ce5f3,open,unclassified,mathematical physics / math-ph,"Problem 5.9 (Kime action–angle atlas on $\mathcal{O}_{m,s}$ ): Construct an atlas of Darboux charts on $\mathcal{O}_{m,s}$ adapted to the kime fibration, i.e., charts of the form $(x^{i},p_{i};\varphi,S_{z}^{\prime})$ in which the directional factor is the kime cylinder of Lemma 5.2 for the little-group sphere, and quantify the obstruction to a single global chart. The fiber $\mathbb{S}^{2}_{s}$ has $\int_{\mathbb{S}^{2}_{s}}\omega_{s}=4\pi s\neq 0$ , so no global Darboux chart exists, and the minimal atlas is governed by the class $[\omega_{s}]/2\pi\hbar$ , which is integral iff $2s/\hbar\in\mathbb{Z}$ (Weil integrality; ). Make precise, within Assumption 1.1 , the resulting statement that a consistent single-valued kime phase law on the directional fiber exists iff the spin is (half-)integer in units of $\hbar$ , the sharpest available classical bridge to spin- $\tfrac{1}{2}$ , and the direct analogue for Problem (III) of the $2\pi$ -holonomy correction anticipated in Problem 3.12 .","enumerated in arXiv paper titled 'open problems' (formal problem environment): Kime-Representation Formulations of Three Open Problems in the Foundations of Classical Mechanics: Uncertainty, Invarian",,https://arxiv.org/abs/2607.07851,
ax-2607.07851-ef7fded5,open,unclassified,mathematical physics / math-ph,"Problem 4.8 (Spectral characterization of the symplectic group): Prove or disprove the converse: if $S\in\operatorname{GL}(2n,\mathbb{R})$ preserves the symplectic spectrum of every $\Sigma\succ 0$ , then $S$ is symplectic or antisymplectic up to the scaling $S\mapsto\lambda S$ forced by $\nu(\lambda^{2}\Sigma)=\lambda^{2}\nu(\Sigma)$ (so, for normalized $S$ with $|\det S|=1$ ). A proof would characterize $\operatorname{Sp}(2n,\mathbb{R})$ purely by an estimable statistical invariant (symplectic spectra of covariance matrices), replacing the geometric definition by an information-theoretic one, the sharpest available answer to Problem (II)’s request for an entropy-first derivation of the symplectic structure.","enumerated in arXiv paper titled 'open problems' (formal problem environment): Kime-Representation Formulations of Three Open Problems in the Foundations of Classical Mechanics: Uncertainty, Invarian",,https://arxiv.org/abs/2607.07851,
ax-2607.07851-f2b5993d,open,unclassified,mathematical physics / math-ph,"Problem 3.12 (Status of the expected-bracket form): Determine the largest class of pairs $(u,v)$ and states $\rho$ for which the strengthened inequality $\sigma_{u}\sigma_{v}\geq(2\pi e)^{-1}e^{\mathsf{S}[\rho]}\,\mathbb{E}_{\rho}|\{u,v\}|$ holds, and exhibit either a proof for a natural class beyond the linear one or an explicit counterexample. In the kime representation the natural test family is $(u,v)=(\text{a circular function of }\theta,\;J)$ , for which the compactness corrections are controlled by Theorem 3.7 . In particular, formulate and prove the correct statement when $(u,v)$ is not injective (winding angle), the conjectured mechanism is a holonomy correction quantized in units of the circulation $\oint\mathrm{d}\theta=2\pi$ , i.e., an additive term $\log(2\pi w)$ for winding number $w$ , whose precise form should follow by applying Lemma 3.10 on a fundamental domain and Lemma 3.4 on the quotient.","enumerated in arXiv paper titled 'open problems' (formal problem environment): Kime-Representation Formulations of Three Open Problems in the Foundations of Classical Mechanics: Uncertainty, Invarian",,https://arxiv.org/abs/2607.07851,
ax-2607.12870-25121477,open,needs-theory,computer science / cs.IT,"Problem 6 (Lattice Isomorphism Problem (LIP)): Given two lattices $\Lambda_{1},\Lambda_{2}$ , find (if it exists) an orthogonal matrix $O$ such that $\Lambda_{1}=\Lambda_{2}O.$",enumerated in arXiv paper titled 'open problems' (formal problem environment): A Survey on Code Equivalence: The State-of-the-Art and Open Questions,,https://arxiv.org/abs/2607.12870,
ax-2607.12870-2572e279,open,needs-theory,computer science / cs.IT,"Problem 2 (PEP): Given two $[n,k]_{q}$ linear codes $\mathcal{C},\mathcal{C}^{\prime}$ find (if it exists) a $\sigma\in S_{n}$ such that $\sigma(\mathcal{C})=\mathcal{C}^{\prime}.$",enumerated in arXiv paper titled 'open problems' (formal problem environment): A Survey on Code Equivalence: The State-of-the-Art and Open Questions,,https://arxiv.org/abs/2607.12870,
ax-2607.12870-61f8dff1,open,needs-theory,computer science / cs.IT,"Problem 5 (Weighted Graph Isomorphism Problem): Given $\mathcal{G},\mathcal{G}^{\prime}$ , find (if it exists) isomorphism $\sigma\in\mathcal{S}_{n}$ mapping $\mathcal{G}$ to $\mathcal{G}^{\prime}.$",enumerated in arXiv paper titled 'open problems' (formal problem environment): A Survey on Code Equivalence: The State-of-the-Art and Open Questions,,https://arxiv.org/abs/2607.12870,
ax-2607.12870-c89253d6,open,needs-theory,computer science / cs.IT,"Problem 3 (LEP( $U$ )): Let $U\subseteq\mathbb{F}_{q}^{\star}$ be a multiplicative subgroup. Given two $[n,k]_{q}$ linear codes $\mathcal{C}$ and $\mathcal{C}^{\prime}$ find (if it exists) an isometry $\varphi=(d,\sigma)\in U^{n}\rtimes S_{n}$ such that $\varphi(\mathcal{C})=\mathcal{C}^{\prime}$ .",enumerated in arXiv paper titled 'open problems' (formal problem environment): A Survey on Code Equivalence: The State-of-the-Art and Open Questions,,https://arxiv.org/abs/2607.12870,
ax-2607.12870-d9d3be17,open,needs-theory,computer science / cs.IT,"Problem 1 (LEP): Given two $[n,k]_{q}$ linear codes $\mathcal{C},\mathcal{C}^{\prime}$ find (if it exists) a $\varphi\in(\mathbb{F}_{q}^{\star})^{n}\rtimes S_{n}$ such that $\varphi(\mathcal{C})=\mathcal{C}^{\prime}.$",enumerated in arXiv paper titled 'open problems' (formal problem environment): A Survey on Code Equivalence: The State-of-the-Art and Open Questions,,https://arxiv.org/abs/2607.12870,
ax-2608.01206-609726bd,open,needs-theory,mathematics / math.CO,"Conjecture 1.1 (Caro and Provstgaard ): Let $r>q\geq 0$ , and let $\ell\geq 2$ be even. Then $R(S(r,q,\ell),\mathbb{Z}_{2})=\begin{cases}r\ell+1,&q=0,\\[2.84526pt] (r-q)\ell+q,&q\geq 1.\end{cases}$",enumerated in arXiv paper titled 'open problems' (formal problem environment): On three open problems in zero-sum Ramsey numbers,,https://arxiv.org/abs/2608.01206,
ax-2608.01206-92364c74,open,compute-checkable-small-cases,mathematics / math.CO,"Problem 1.5 (Caro ): Does there exist a tree $T$ with $m$ edges such that $R(T,\mathbb{Z}_{m})>R(T,2)$ ?",enumerated in arXiv paper titled 'open problems' (formal problem environment): On three open problems in zero-sum Ramsey numbers,,https://arxiv.org/abs/2608.01206,
ax-2608.01206-f63280f7,open,compute-checkable-small-cases,mathematics / math.CO,"Conjecture 1.3 (Caro ): Let $F$ be a forest with $m$ edges. Then $R(tF,\mathbb{Z}_{mt})=R(tF,2)$ for all sufficiently large $t$ .",enumerated in arXiv paper titled 'open problems' (formal problem environment): On three open problems in zero-sum Ramsey numbers,,https://arxiv.org/abs/2608.01206,
ax-2608.10119-b711cff3,open,needs-theory,computer science / cs.PF,"The Open Problem: Is $\sigma_{i}$ Derivable, or Calibrated? (Section 4 isolates the paper’s actual unresolved question precisely: $\Gamma$ ’s recursive structure and its two governing conditions (Equations 1 and 2 ) are not in dispute by the evidence gathered so far, but $\sigma_{i}$ is a free parameter in that structure, and only one of the four rows in Table 1 used a $\sigma_{i}$ that was not fit to a specific machine after the fact – and that one row is…)","enumerated in arXiv paper titled 'open problems' (section headings phrased as questions): Machine Shape and Hierarchical Blocking: A Mathematics of Arrays Formalization, with an Open Problem in Hierarchical Sha",,https://arxiv.org/abs/2608.10119,
ax-2608.10119-de68385b,open,needs-theory,computer science / cs.PF,"Testing Transferability: Does $\eta(P)$ Have One Shape? (The most useful question about $\eta(P)$ is not its value on any one machine but whether its shape , once normalized against each machine’s own single-thread baseline, is shared across machines – which would make it a transferable term rather than one requiring fresh calibration every time. This is directly testable with data already in hand: identical calibration sweeps exist for M1 Pro,…)","enumerated in arXiv paper titled 'open problems' (section headings phrased as questions): Machine Shape and Hierarchical Blocking: A Mathematics of Arrays Formalization, with an Open Problem in Hierarchical Sha",,https://arxiv.org/abs/2608.10119,
ax-2608.26628-2d6bf100,open,compute-checkable-small-cases,mathematics / math.LO,Problem 2.1: Let $M$ be an $\omega$ -categorical structure in a first order finite language. Is the group $\operatorname{Out}(G)$ finite?,enumerated in arXiv paper titled 'open problems' (formal problem environment): Open Problems in Mathematical Logic,,https://arxiv.org/abs/2608.26628,
ax-2608.26628-4ac787b4,open,needs-theory,mathematics / math.LO,"Problem 9.1: Does this hold with a larger degree notion, for example, LR-degrees?",enumerated in arXiv paper titled 'open problems' (formal problem environment): Open Problems in Mathematical Logic,,https://arxiv.org/abs/2608.26628,
ax-2608.26628-52705d9a,open,compute-checkable-small-cases,mathematics / math.LO,Problem 11.1: Does there exist a complete first-order theory $T$ in a countable language such that $=\,enumerated in arXiv paper titled 'open problems' (formal problem environment): Open Problems in Mathematical Logic,,https://arxiv.org/abs/2608.26628,
ax-2608.26628-57a317a1,open,needs-theory,mathematics / math.LO,"Problem 6.1: For vector space over fixed $\mathbb{K}$ , is choose a basis require the full Axiom of Choice?",enumerated in arXiv paper titled 'open problems' (formal problem environment): Open Problems in Mathematical Logic,,https://arxiv.org/abs/2608.26628,
ax-2608.26628-57d77062,open,needs-theory,mathematics / math.LO,Problem 8.2 (): Is $\thicksim$ hyperfinite?,enumerated in arXiv paper titled 'open problems' (formal problem environment): Open Problems in Mathematical Logic,,https://arxiv.org/abs/2608.26628,
ax-2608.26628-5891ca80,open,compute-checkable-small-cases,mathematics / math.LO,"Problem 4.1: Is there a Kakeya set $A\subseteq\mathbb{R}^{2}$ such that $x\oplus y\geq_{T}\emptyset^{\prime}$ for every $(x,y)\in A$ ?",enumerated in arXiv paper titled 'open problems' (formal problem environment): Open Problems in Mathematical Logic,,https://arxiv.org/abs/2608.26628,
ax-2608.26628-5b6a141d,open,compute-checkable-small-cases,mathematics / math.LO,"Problem 1.4: Toward a negative answer to Levin’s Problem 1.2, are the following true? (1) Are there $2$ -random reals $x $x^{\prime}\equiv_{T}z^{\prime}?$ (2) Are there $2$ -random reals $x $\left|K(x\upharpoonright n)-K(z\upharpoonright n)\right|=O(1)?$ (3) Is there a Turing ideal $B$ generated by a chain of $2$ -random reals such that $B$ contains no $x$ -random real for any $x\in B$ ? (4) What are the jumps of $2$ -random reals? See .",enumerated in arXiv paper titled 'open problems' (formal problem environment): Open Problems in Mathematical Logic,,https://arxiv.org/abs/2608.26628,
ax-2608.26628-5f7b34d2,open,needs-theory,mathematics / math.LO,Problem 3.1: Determine the equivalence classes of quantifier patterns at the fourth level of the arithmetical hierarchy: (1) How many $\equiv_{m}$ -equivalence classes of $\Sigma^{0}_{4}$ quantifier patterns are there? (2) How many $\equiv_{m}$ -equivalence classes of $\Pi^{0}_{4}$ quantifier patterns are there? (3) How many $\equiv_{dm}$ -equivalence classes of $\Sigma^{0}_{4}$ quantifier patterns are there?,enumerated in arXiv paper titled 'open problems' (formal problem environment): Open Problems in Mathematical Logic,,https://arxiv.org/abs/2608.26628,
ax-2608.26628-6994e84c,open,compute-checkable-small-cases,mathematics / math.LO,"Problem 1.2 (Levin): Is it true that, for every $X\subseteq 2^{\omega}$ with $\mu(X)=1$ , there exists $Y\subseteq 2^{\omega}$ with $\mu(Y)=1$ such that, for every chain $B\subseteq Y$ , there is some $\alpha\in X$ satisfying $[B]=[\alpha]$ ?",enumerated in arXiv paper titled 'open problems' (formal problem environment): Open Problems in Mathematical Logic,,https://arxiv.org/abs/2608.26628,
ax-2608.26628-7704b05b,open,compute-checkable-small-cases,mathematics / math.LO,"Problem 3.2: Is there a decision procedure which, given quantifier patterns $\overline{P},\overline{Q}\in\{\exists,\forall,\exists^{\infty},\forall^{\infty}\}^{ decides whether $\langle\overline{P}\rangle\leq_{m}\langle\overline{Q}\rangle?$",enumerated in arXiv paper titled 'open problems' (formal problem environment): Open Problems in Mathematical Logic,,https://arxiv.org/abs/2608.26628,
ax-2608.26628-86221a07,open,compute-checkable-small-cases,mathematics / math.LO,Problem 1.3: Is there a chain $B\subseteq\mathrm{ML}(\emptyset^{\prime})$ such that $[B]\neq[\alpha]$ for every $\alpha\in\mathrm{ML}(\emptyset^{\prime})$ ?,enumerated in arXiv paper titled 'open problems' (formal problem environment): Open Problems in Mathematical Logic,,https://arxiv.org/abs/2608.26628,
ax-2608.26628-8a2d61ce,open,needs-theory,mathematics / math.LO,Problem 4.2: Does every Borel Kakeya set $A$ contain a minimal Kakeya set $B\subseteq A$ ?,enumerated in arXiv paper titled 'open problems' (formal problem environment): Open Problems in Mathematical Logic,,https://arxiv.org/abs/2608.26628,
ax-2608.26628-8c099010,open,compute-checkable-small-cases,mathematics / math.LO,Problem 7.1: Is there a Davies example provably in ZFC?,enumerated in arXiv paper titled 'open problems' (formal problem environment): Open Problems in Mathematical Logic,,https://arxiv.org/abs/2608.26628,
ax-2608.26628-a4619842,open,needs-theory,mathematics / math.LO,"Problem 10.1: If $L_{\beta}\models\mbox{$\mathit{V}=\mathit{HC}$}$ , then $\displaystyle\left(\mathcal{P}(\omega)\cap L_{\beta}\models\bm{\Pi}_{m+1}^{1}\mathsf{-MI}\,\right)\iff\beta\mbox{ supports a strong }\Sigma_{m+1}\textrm{-nesting}.$",enumerated in arXiv paper titled 'open problems' (formal problem environment): Open Problems in Mathematical Logic,,https://arxiv.org/abs/2608.26628,
ax-2608.26628-a6eeb53e,open,needs-theory,mathematics / math.LO,Problem 8.1: Find more models $M$ and forcing notions $\mathbb{P}\in M$ such that $(*)$ is true.,enumerated in arXiv paper titled 'open problems' (formal problem environment): Open Problems in Mathematical Logic,,https://arxiv.org/abs/2608.26628,
ax-2608.26628-c1ab844e,open,compute-checkable-small-cases,mathematics / math.LO,"Problem 1.5: Is there an injective one-way function on the reals? More precisely, is there a partial computable real function $f$ such that (i) $f$ is injective and preserves randomness, (ii) the domain of $f$ has positive Lebesgue measure, and (iii) $f(x)",enumerated in arXiv paper titled 'open problems' (formal problem environment): Open Problems in Mathematical Logic,,https://arxiv.org/abs/2608.26628,
ax-2608.26628-ce555a8c,open,compute-checkable-small-cases,mathematics / math.LO,Problem 5.1: Is there a Borel two-point set?,enumerated in arXiv paper titled 'open problems' (formal problem environment): Open Problems in Mathematical Logic,,https://arxiv.org/abs/2608.26628,
ax-2608.26628-d05ec186,open,needs-theory,mathematics / math.LO,Problem 2.2: Does $\leq_{SJT}$ imply $\leq_{LR}$ ?,enumerated in arXiv paper titled 'open problems' (formal problem environment): Open Problems in Mathematical Logic,,https://arxiv.org/abs/2608.26628,
ax-2608.26628-d38ef033,open,compute-checkable-small-cases,mathematics / math.LO,Problem 2.3: Is there a $\Pi^{0}_{1}$ class $C$ such that $|C\cap{\rm Rec}|=\infty$ and $C\cap{\rm Rec}$ is maximally almost disjoint among the recursive sets?,enumerated in arXiv paper titled 'open problems' (formal problem environment): Open Problems in Mathematical Logic,,https://arxiv.org/abs/2608.26628,
ax-2608.26628-d66e4864,open,needs-theory,mathematics / math.LO,Problem 5.2 (Montalbán–Nies ): Does every Borel field have a Borel algebraically closed extension?,enumerated in arXiv paper titled 'open problems' (formal problem environment): Open Problems in Mathematical Logic,,https://arxiv.org/abs/2608.26628,
ax-2608.26628-e24ebfed,open,needs-theory,mathematics / math.LO,"Problem 1.1 (Levin, private communications): Are the following true? (1) Can every chain of random reals in the Turing degrees be combined into a single random real? (2) Can every Turing ideal generated by random reals be generated by the columns of a single random real?",enumerated in arXiv paper titled 'open problems' (formal problem environment): Open Problems in Mathematical Logic,,https://arxiv.org/abs/2608.26628,
ax-2608.26628-e5ad5fbc,open,needs-theory,mathematics / math.LO,Problem 8.3: Suppose there are a model $M$ and a forcing notion $\mathbb{P}\in M$ such that $(*)$ is true. Can we deduce that $M\models[\exists\ \text{a large cardinal}]$ ?,enumerated in arXiv paper titled 'open problems' (formal problem environment): Open Problems in Mathematical Logic,,https://arxiv.org/abs/2608.26628,
cc-z1-listwise-collapse-global-discrimination,open,definition-dispute,CS / ML agents,"Contested in our registry (1 supporting, 1 refuting): The drop of Accuracy@1 from 61.3% (N=2) to 31.1% (N=5) in Table 3 indicates that the LLM judge lacks global discrimination capability beyond binary interactions.",contested claim,Read the two evidence spans side by side and decide whether they disagree about the claim's meaning (then split the claim) or about the numbers (then re-run the cheapest computation).,,
erdos-1,open,needs-theory,mathematics / number theory,"If $A\subseteq \{1,\ldots,N\}$ with $\lvert A\rvert=n$ is such that the subset sums $\sum_{a\in S}a$ are distinct for all $S\subseteq A$ then\[N \gg 2^{n}.\]","Erdős problems database (status: open; prize: $500; tags: number theory, additive combinatorics)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A276661 and check the conjectured growth against them.,https://www.erdosproblems.com/1,
erdos-10,open,needs-theory,mathematics / number theory,Is there some $k$ such that every large integer is the sum of a prime and at most $k$ powers of 2?,"Erdős problems database (status: open; prize: no; tags: number theory, additive basis, primes)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A387053 and check the conjectured growth against them.,https://www.erdosproblems.com/10,
erdos-100,open,needs-theory,mathematics / geometry,Let $A$ be a set of $n$ points in $\mathbb{R}^2$ such that all pairwise distances are at least $1$ and if two distinct distances differ then they differ by at least $1$. Is the diameter of $A$ $\gg n$?,"Erdős problems database (status: open; prize: no; tags: geometry, distances)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/100,
erdos-1002,open,needs-theory,mathematics / analysis,"For any $0<\alpha<1$, let\[f(\alpha,n)=\frac{1}{\log n}\sum_{1\leq k\leq n}(\tfrac{1}{2}-\{ \alpha k\}).\]Does $f(\alpha,n)$ have an asymptotic distribution function? In other words, is there a non-decreasing function $g$ such that $g(-\infty)=0$, $g(\infty)=1$, and\[\lim_{n\to \infty}\lvert \{ \alpha\in (0,1): f(\alpha,n)\leq c\}\rvert=g(c)?\]","Erdős problems database (status: open; prize: no; tags: analysis, diophantine approximation)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1002,
erdos-1003,open,needs-theory,mathematics / number theory,"Are there infinitely many solutions to $\phi(n)=\phi(n+1)$, where $\phi$ is the Euler totient function?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A001274 and check the conjectured growth against them.,https://www.erdosproblems.com/1003,
erdos-1004,open,needs-theory,mathematics / number theory,"Let $c>0$. If $x$ is sufficiently large then does there exist $n\leq x$ such that the values of $\phi(n+k)$ are all distinct for $1\leq k\leq (\log x)^c$, where $\phi$ is the Euler totient function?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1004,
erdos-101,open,needs-theory,mathematics / geometry,"Given $n$ points in $\mathbb{R}^2$, no five of which are on a line, the number of lines containing four points is $o(n^2)$.",Erdős problems database (status: open; prize: $100; tags: geometry),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A006065, possible and check the conjectured growth against them.",https://www.erdosproblems.com/101,
erdos-1011,open,needs-theory,mathematics / graph theory,Let $f_r(n)$ be minimal such that every graph on $n$ vertices with $\geq f_r(n)$ edges and chromatic number $\geq r$ contains a triangle. Determine $f_r(n)$.,Erdős problems database (status: open; prize: no; tags: graph theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1011,
erdos-1013,open,needs-theory,mathematics / graph theory,"Let $h_3(k)$ be the minimal $n$ such that there exists a triangle-free graph on $n$ vertices with chromatic number $k$. Find an asymptotic for $h_3(k)$, and also prove\[\lim_{k\to \infty}\frac{h_3(k+1)}{h_3(k)}=1.\]",Erdős problems database (status: open; prize: no; tags: graph theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A292528 and check the conjectured growth against them.,https://www.erdosproblems.com/1013,
erdos-1016,open,needs-theory,mathematics / graph theory,"Let $h(n)$ be minimal such that there is a graph on $n$ vertices with $n+h(n)$ edges which contains a cycle on $k$ vertices, for all $3\leq k\leq n$. Estimate $h(n)$. In particular, is it true that\[h(n) \geq \log_2n+\log_*n-O(1),\]where $\log_*n$ is the iterated logarithmic function?","Erdős problems database (status: open; prize: no; tags: graph theory, cycles)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A105206 and check the conjectured growth against them.,https://www.erdosproblems.com/1016,
erdos-1017,open,needs-theory,mathematics / graph theory,"Let $f(n,k)$ be such that every graph on $n$ vertices and $k$ edges can be partitioned into at most $f(n,k)$ edge-disjoint complete graphs. Estimate $f(n,k)$ for $k>n^2/4$.",Erdős problems database (status: open; prize: no; tags: graph theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1017,
erdos-102,open,needs-theory,mathematics / geometry,"Let $c>0$ and $h_c(n)$ be such that for any $n$ points in $\mathbb{R}^2$ such that there are $\geq cn^2$ lines each containing more than three points, there must be some line containing $h_c(n)$ many points. Estimate $h_c(n)$. Is it true that, for fixed $c>0$, we have $h_c(n)\to \infty$?",Erdős problems database (status: open; prize: no; tags: geometry),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/102,
erdos-1020,open,compute-checkable-small-cases,mathematics / graph theory,"Open, but could be disproved with a finite counterexample. Let $f(n;r,k)$ be the maximal number of edges in an $r$-uniform hypergraph which contains no set of $k$ many independent edges. For all $r\geq 3$,\[f(n;r,k)=\max\left(\binom{rk-1}{r}, \binom{n}{r}-\binom{n-k+1}{r}\right).\]","Erdős problems database (status: falsifiable; prize: no; tags: graph theory, hypergraphs)",The database marks this as refutable by a finite computation if false: search for a counterexample. Extend OEIS N/A first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/1020,
erdos-1029,open,needs-theory,mathematics / graph theory,"If $R(k)$ is the Ramsey number for $K_k$, the minimal $n$ such that every $2$-colouring of the edges of $K_n$ contains a monochromatic copy of $K_k$, then\[\frac{R(k)}{k2^{k/2}}\to \infty.\]","Erdős problems database (status: open; prize: $100; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A059442 and check the conjectured growth against them.,https://www.erdosproblems.com/1029,
erdos-103,open,needs-theory,mathematics / geometry,"Let $h(n)$ count the number of incongruent sets of $n$ points in $\mathbb{R}^2$ which minimise the diameter subject to the constraint that $d(x,y)\geq 1$ for all points $x\neq y$. Is it true that $h(n)\to \infty$?","Erdős problems database (status: open; prize: no; tags: geometry, distances)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/103,
erdos-1030,open,needs-theory,mathematics / graph theory,"Let $R(k,l)$ be the usual Ramsey number: the smallest $n$ such that if the edges of $K_n$ are coloured red and blue then there exists either a red $K_k$ or a blue $K_l$. Prove the existence of some $c>0$ such that\[\lim_{k\to \infty}\frac{R(k+1,k)}{R(k,k)}> 1+c.\]","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A000791, A059442 and check the conjectured growth against them.",https://www.erdosproblems.com/1030,
erdos-1032,open,needs-theory,mathematics / graph theory,"We say that a graph is $4$-chromatic critical if it has chromatic number $4$, and removing any edge decreases the chromatic number to $3$. Is there, for arbitrarily large $n$, a $4$-chromatic critical graph on $n$ vertices with minimum degree $\gg n$?","Erdős problems database (status: open; prize: no; tags: graph theory, chromatic number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1032,
erdos-1033,open,needs-theory,mathematics / graph theory,"Let $h(n)$ be such that every graph on $n$ vertices with $>n^2/4$ many edges contains a triangle whose vertices have degrees summing to at least $h(n)$. Estimate $h(n)$. In particular, is it true that\[h(n)\geq (2(\sqrt{3}-1)-o(1))n?\]",Erdős problems database (status: open; prize: no; tags: graph theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1033,
erdos-1035,open,needs-theory,mathematics / graph theory,Is there a constant $c>0$ such that every graph on $2^n$ vertices with minimum degree $>(1-c)2^n$ contains the $n$-dimensional hypercube $Q_n$?,Erdős problems database (status: open; prize: no; tags: graph theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1035,
erdos-1038,open,needs-theory,mathematics / analysis,"Determine the infimum and supremum of\[\lvert \{ x\in \mathbb{R} : \lvert f(x)\rvert < 1\}\rvert\]as $f\in \mathbb{R}[x]$ ranges over all non-constant monic polynomials, all of whose roots are real and in the interval $[-1,1]$.",Erdős problems database (status: open; prize: no; tags: analysis),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1038,
erdos-1039,open,needs-theory,mathematics / analysis,"Let $f(z)=\prod_{i=1}^n(z-z_i)\in \mathbb{C}[z]$ with $\lvert z_i\rvert \leq 1$ for all $i$. Let $\rho(f)$ be the radius of the largest disc which is contained in $\{z: \lvert f(z)\rvert< 1\}$. Determine the behaviour of $\rho(f)$. In particular, is it always true that $\rho(f)\gg 1/n$?",Erdős problems database (status: open; prize: no; tags: analysis),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1039,
erdos-104,open,needs-theory,mathematics / geometry,Given $n$ points in $\mathbb{R}^2$ the number of distinct unit circles containing at least three points is $o(n^2)$.,Erdős problems database (status: open; prize: $100; tags: geometry),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A003829 and check the conjectured growth against them.,https://www.erdosproblems.com/104,
erdos-1040,open,needs-theory,mathematics / analysis,"Let $F\subseteq \mathbb{C}$ be a closed infinite set, and let $\mu(F)$ be the infimum of\[\lvert \{ z: \lvert f(z)\rvert < 1\}\rvert,\]as $f$ ranges over all polynomials of the shape $\prod (z-z_i)$ with $z_i\in F$. Is $\mu(F)$ determined by the transfinite diameter of $F$? In particular, is $\mu(F)=0$ whenever the transfinite diameter of $F$ is $\geq 1$?",Erdős problems database (status: open; prize: no; tags: analysis),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1040,
erdos-1041,open,compute-checkable-small-cases,mathematics / analysis,"Open, but could be disproved with a finite counterexample. Let $f(z)=\prod_{i=1}^n(z-z_i)\in \mathbb{C}[z]$ with $\lvert z_i\rvert < 1$ for all $i$. Must there always exist a path of length less than $2$ in\[\{z: \lvert f(z)\rvert < 1\}\]which connects two of the roots of $f$?",Erdős problems database (status: falsifiable; prize: no; tags: analysis),The database marks this as refutable by a finite computation if false: search for a counterexample. Extend OEIS N/A first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/1041,
erdos-1045,open,needs-theory,mathematics / analysis,"Let $z_1,\ldots,z_n\in \mathbb{C}$ with $\lvert z_i-z_j\rvert\leq 2$ for all $i,j$, and\[\Delta(z_1,\ldots,z_n)=\prod_{i\neq j}\lvert z_i-z_j\rvert.\]What is the maximum possible value of $\Delta$? Is it maximised by taking the $z_i$ to be the vertices of a regular polygon?",Erdős problems database (status: open; prize: no; tags: analysis),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1045,
erdos-1049,open,needs-theory,mathematics / irrationality,"Let $t>1$ be a rational number. Is\[\sum_{n=1}^\infty\frac{1}{t^n-1}=\sum_{n=1}^\infty \frac{\tau(n)}{t^n}\]irrational, where $\tau(n)$ counts the divisors of $n$?",Erdős problems database (status: open; prize: no; tags: irrationality),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1049,
erdos-1052,open,needs-theory,mathematics / number theory,"A unitary divisor of $n$ is $d\mid n$ such that $(d,n/d)=1$. A number $n\geq 1$ is a unitary perfect number if it is the sum of its unitary divisors (aside from $n$ itself). Are there only finite many unitary perfect numbers?",Erdős problems database (status: open; prize: $10; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A002827 and check the conjectured growth against them.,https://www.erdosproblems.com/1052,
erdos-1053,open,needs-theory,mathematics / number theory,"Call a number $k$-perfect if $\sigma(n)=kn$, where $\sigma(n)$ is the sum of the divisors of $n$. Must $k=o(\log\log n)$?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A007539 and check the conjectured growth against them.,https://www.erdosproblems.com/1053,
erdos-1054,open,needs-theory,mathematics / number theory,"Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $f(n)=o(n)$? Or is this true only for almost all $n$, and $\limsup f(n)/n=\infty$?","Erdős problems database (status: open; prize: no; tags: number theory, divisors)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A167485 and check the conjectured growth against them.,https://www.erdosproblems.com/1054,
erdos-1055,open,needs-theory,mathematics / number theory,"A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. Are there infinitely many primes in each class? If $p_r$ is the least prime in class $r$, then how does $p_r^{1/r}$ behave?","Erdős problems database (status: open; prize: no; tags: number theory, primes)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A005113 and check the conjectured growth against them.,https://www.erdosproblems.com/1055,
erdos-1056,open,needs-theory,mathematics / number theory,"Let $k\geq 2$. Does there exist a prime $p$ and consecutive intervals $I_1,\ldots,I_k$ such that\[\prod_{n\in I_i}n \equiv 1\pmod{p}\]for all $1\leq i\leq k$?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A060427 and check the conjectured growth against them.,https://www.erdosproblems.com/1056,
erdos-1057,open,needs-theory,mathematics / number theory,"Let $C(x)$ count the number of Carmichael numbers in the interval $[1,x]$. Is it true that $C(x)=x^{1-o(1)}$?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A006931 and check the conjectured growth against them.,https://www.erdosproblems.com/1057,
erdos-1059,open,needs-theory,mathematics / number theory,Are there infinitely many primes $p$ such that $p-k!$ is composite for each $k$ such that $1\leq k!<p$?,"Erdős problems database (status: open; prize: no; tags: number theory, primes)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A064152 and check the conjectured growth against them.,https://www.erdosproblems.com/1059,
erdos-1060,open,needs-theory,mathematics / number theory,"Let $f(n)$ count the number of solutions to $k\sigma(k)=n$, where $\sigma(k)$ is the sum of divisors of $k$. Is it true that $f(n)\leq n^{o(\frac{1}{\log\log n})}$? Perhaps even $\leq (\log n)^{O(1)}$?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A327153 and check the conjectured growth against them.,https://www.erdosproblems.com/1060,
erdos-1061,open,needs-theory,mathematics / number theory,"How many solutions are there to\[\sigma(a)+\sigma(b)=\sigma(a+b)\]with $a+b\leq x$, where $\sigma$ is the sum of divisors function? Is it $\sim cx$ for some constant $c>0$?",Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A110177, possible and check the conjectured growth against them.",https://www.erdosproblems.com/1061,
erdos-1062,open,needs-theory,mathematics / number theory,"Let $f(n)$ be the size of the largest subset $A\subseteq \{1,\ldots,n\}$ such that there are no three distinct elements $a,b,c\in A$ such that $a\mid b$ and $a\mid c$. How large can $f(n)$ be? Is $\lim f(n)/n$ irrational?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A038372 and check the conjectured growth against them.,https://www.erdosproblems.com/1062,
erdos-1063,open,needs-theory,mathematics / number theory,Let $k\geq 2$ and define $n_k\geq 2k$ to be the least value of $n$ such that $n-i$ divides $\binom{n}{k}$ for all but one $0\leq i<k$. Estimate $n_k$.,Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A389360 and check the conjectured growth against them.,https://www.erdosproblems.com/1063,
erdos-1065,open,needs-theory,mathematics / number theory,Are there infinitely many primes $p$ such that $p=2^kq+1$ for some prime $q$ and $k\geq 0$? Or $p=2^k3^lq+1$?,Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A074781, A339465 and check the conjectured growth against them.",https://www.erdosproblems.com/1065,
erdos-1066,open,needs-theory,mathematics / graph theory,"Let $G$ be a graph given by $n$ points in $\mathbb{R}^2$, where any two distinct points are at least distance $1$ apart, and we draw an edge between two points if they are distance $1$ apart. Let $g(n)$ be maximal such that any such graph always has an independent set on at least $g(n)$ vertices. Estimate $g(n)$, or perhaps $\lim \frac{g(n)}{n}$.","Erdős problems database (status: open; prize: no; tags: graph theory, planar graphs)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1066,
erdos-1068,open,needs-theory,mathematics / graph theory,Does every graph with chromatic number $\aleph_1$ contain a countable subgraph which is infinitely vertex-connected?,"Erdős problems database (status: open; prize: no; tags: graph theory, set theory, chromatic number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1068,
erdos-107,open,compute-checkable-small-cases,mathematics / geometry,"Open, but could be disproved with a finite counterexample. - $500 Let $f(n)$ be minimal such that any $f(n)$ points in $\mathbb{R}^2$, no three on a line, contain $n$ points which form the vertices of a convex $n$-gon. Prove that $f(n)=2^{n-2}+1$.","Erdős problems database (status: falsifiable; prize: $500; tags: geometry, convex)",The database marks this as refutable by a finite computation if false: search for a counterexample. Extend OEIS A000051 first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/107,
erdos-1070,open,needs-theory,mathematics / geometry,"Let $f(n)$ be maximal such that, given any $n$ points in $\mathbb{R}^2$, there exist $f(n)$ points such that no two are distance $1$ apart. Estimate $f(n)$. In particular, is it true that $f(n)\geq n/4$?",Erdős problems database (status: open; prize: no; tags: geometry),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1070,
erdos-1072,open,needs-theory,mathematics / number theory,"For any prime $p$, let $f(p)$ be the least integer such that $f(p)!+1\equiv 0\pmod{p}$. Is it true that there are infinitely many $p$ for which $f(p)=p-1$? Is it true that $f(p)/p\to 0$ for almost all $p$?",Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A073944, A072937, A154554 and check the conjectured growth against them.",https://www.erdosproblems.com/1072,
erdos-1073,open,needs-theory,mathematics / number theory,Let $A(x)$ count the number of composite $u<x$ such that $n!+1\equiv 0\pmod{u}$ for some $n$. Is it true that $A(x)\leq x^{o(1)}$?,Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A256519 and check the conjectured growth against them.,https://www.erdosproblems.com/1073,
erdos-1074,open,needs-theory,mathematics / number theory,"Let $S$ be the set of all $m\geq 1$ such that there exists a prime $p\not\equiv 1\pmod{m}$ such that $m!+1\equiv 0\pmod{p}$. Does\[\lim \frac{\lvert S\cap [1,x]\rvert}{x}\]exist? What is it? Similarly, if $P$ is the set of all primes $p$ such that there exists an $m$ with $p\not\equiv 1\pmod{m}$ such that $m!+1\equiv 0\pmod{p}$, then does\[\lim \frac{\lvert P\cap [1,x]\rvert}{\pi(x)}\]exist? What is it?",Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A063980, A064164 and check the conjectured growth against them.",https://www.erdosproblems.com/1074,
erdos-1075,open,needs-theory,mathematics / hypergraphs,"Let $r\geq 3$. There exists $c_r>r^{-r}$ such that, for any $\epsilon>0$, if $n$ is sufficiently large, the following holds. Any $r$-uniform hypergraph on $n$ vertices with at least $(1+\epsilon)(n/r)^r$ many edges contains a subgraph on $m$ vertices with at least $c_rm^r$ edges, where $m=m(n)\to \infty$ as $n\to \infty$.",Erdős problems database (status: open; prize: no; tags: hypergraphs),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1075,
erdos-108,open,needs-theory,mathematics / graph theory,"For every $r\geq 4$ and $k\geq 2$ is there some finite $f(k,r)$ such that every graph of chromatic number $\geq f(k,r)$ contains a subgraph of girth $\geq r$ and chromatic number $\geq k$?","Erdős problems database (status: open; prize: no; tags: graph theory, chromatic number, cycles)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/108,
erdos-1082,open,compute-checkable-small-cases,mathematics / geometry,"Open, but could be disproved with a finite counterexample. Let $A\subset \mathbb{R}^2$ be a set of $n$ points with no three on a line. Does $A$ determine at least $\lfloor n/2\rfloor$ distinct distances? In fact, must there exist a single point from which there are at least $\lfloor n/2\rfloor$ distinct distances?","Erdős problems database (status: falsifiable; prize: no; tags: geometry, distances)",The database marks this as refutable by a finite computation if false: search for a counterexample. Extend OEIS possible first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/1082,
erdos-1083,open,needs-theory,mathematics / geometry,"Let $d\geq 3$, and let $f_d(n)$ be the minimal $m$ such that every set of $n$ points in $\mathbb{R}^d$ determines at least $m$ distinct distances. Estimate $f_d(n)$ - in particular, is it true that\[f_d(n)=n^{\frac{2}{d}-o(1)}?\]","Erdős problems database (status: open; prize: no; tags: geometry, distances)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A186704, possible and check the conjectured growth against them.",https://www.erdosproblems.com/1083,
erdos-1084,open,needs-theory,mathematics / geometry,"Let $f_d(n)$ be minimal such that in any collection of $n$ points in $\mathbb{R}^d$, all of distance at least $1$ apart, there are at most $f_d(n)$ many pairs of points which are distance $1$ apart. Estimate $f_d(n)$.","Erdős problems database (status: open; prize: no; tags: geometry, distances)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A045945, possible and check the conjectured growth against them.",https://www.erdosproblems.com/1084,
erdos-1085,open,needs-theory,mathematics / geometry,"Let $f_d(n)$ be minimal such that, in any set of $n$ points in $\mathbb{R}^d$, there exist at most $f_d(n)$ pairs of points which distance $1$ apart. Estimate $f_d(n)$.","Erdős problems database (status: open; prize: no; tags: geometry, distances)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A186705, possible and check the conjectured growth against them.",https://www.erdosproblems.com/1085,
erdos-1086,open,needs-theory,mathematics / geometry,Let $g(n)$ be minimal such that any set of $n$ points in $\mathbb{R}^2$ contains the vertices of at most $g(n)$ many triangles with the same area. Estimate $g(n)$.,"Erdős problems database (status: open; prize: no; tags: geometry, distances)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1086,
erdos-1087,open,needs-theory,mathematics / geometry,"Let $f(n)$ be minimal such that every set of $n$ points in $\mathbb{R}^2$ contains at most $f(n)$ many sets of four points which are 'degenerate' in the sense that some pair are the same distance apart. Estimate $f(n)$ - in particular, is it true that $f(n)\leq n^{3+o(1)}$?","Erdős problems database (status: open; prize: no; tags: geometry, distances)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1087,
erdos-1088,open,needs-theory,mathematics / geometry,"Let $f_d(n)$ be the minimal $m$ such that any set of $m$ points in $\mathbb{R}^d$ contains a set of $n$ points such that any two determined distances are distinct. Estimate $f_d(n)$. In particular, is it true that, for fixed $n\geq 3$,\[f_d(n)=2^{o(d)}?\]",Erdős problems database (status: open; prize: no; tags: geometry),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1088,
erdos-1093,open,needs-theory,mathematics / number theory,"For $n\geq 2k$ we define the deficiency of $\binom{n}{k}$ as follows. If $\binom{n}{k}$ is divisible by a prime $p\leq k$ then the deficiency is undefined. Otherwise, the deficiency is the number of $0\leq i<k$ such that $n-i$ is $k$-smooth, that is, divisible only by primes $\leq k$. Are there infinitely many binomial coefficients with deficiency $1$? Are there only finitely many with deficiency $>1$?","Erdős problems database (status: open; prize: no; tags: number theory, binomial coefficients)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1093,
erdos-1094,open,needs-theory,mathematics / number theory,"For all $n\geq 2k$ the least prime factor of $\binom{n}{k}$ is $\leq \max(n/k,k)$, with only finitely many exceptions.","Erdős problems database (status: open; prize: no; tags: number theory, binomial coefficients)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1094,
erdos-1095,open,needs-theory,mathematics / number theory,Let $g(k)>k+1$ be the smallest $n$ such that all prime factors of $\binom{n}{k}$ are $>k$. Estimate $g(k)$.,"Erdős problems database (status: open; prize: no; tags: number theory, binomial coefficients)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A003458 and check the conjectured growth against them.,https://www.erdosproblems.com/1095,
erdos-1097,open,needs-theory,mathematics / number theory,"Let $A$ be a set of $n$ integers. How many distinct $d$ can occur as the common difference of a three-term arithmetic progression in $A$? In particular, are there always $O(n^{3/2})$ many such $d$?","Erdős problems database (status: open; prize: no; tags: number theory, additive combinatorics)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1097,
erdos-11,open,needs-theory,mathematics / number theory,Is every large odd integer $n$ the sum of a squarefree number and a power of 2?,"Erdős problems database (status: open; prize: no; tags: number theory, additive basis)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A001220, A377587 and check the conjectured growth against them.",https://www.erdosproblems.com/11,
erdos-1100,open,needs-theory,mathematics / number theory,"If $1=d_1<\cdots<d_{\tau(n)}=n$ are the divisors of $n$, then let $\tau_\perp(n)$ count the number of $i$ for which $(d_i,d_{i+1})=1$. Is it true that $\tau_\perp(n)/\omega(n)\to \infty$ for almost all $n$? Is it true that\[\tau_\perp(n)< \exp((\log n)^{o(1)})\]for all $n$? Let\[g(k) = \max_{\omega(n)=k}\tau_\perp(n),\]where $\omega(n)$ counts the number of distinct prime divisors of $n$, and $n$ is restricted to squarefree integers. Determine the growth of $g(k)$.","Erdős problems database (status: open; prize: no; tags: number theory, divisors)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A325864, possible and check the conjectured growth against them.",https://www.erdosproblems.com/1100,
erdos-1101,open,needs-theory,mathematics / number theory,"If $u=\{u_1<u_2<\cdots\}$ is a sequence of integers such that $(u_i,u_j)=1$ for all $i\neq j$ and $\sum \frac{1}{u_i}<\infty$ then let $\{a_1<a_2<\cdots\}$ be the sequence of integers which are not divisible by any of the $u_i$. For any $x$ define $t_x$ by\[u_1\cdots u_{t_x}\leq x< u_1\cdots u_{t_x}u_{t_x+1}.\]We call such a sequence $u_i$ good if, for all $\epsilon>0$, if $x$ is sufficiently large then\[\max_{a_k<x} (a_{k+1}-a_k) < (1+\epsilon)t_x \prod_{i}\left(1-\frac{1}{u_i}\right)^{-1}.\]Is there a good sequence such that $u_n< n^{O(1)}$? Is there a good sequence such that $u_n\leq e^{o(n)}$?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1101,
erdos-1103,open,needs-theory,mathematics / number theory,Let $A$ be an infinite sequence of integers such that every $n\in A+A$ is squarefree. How fast must $A$ grow?,Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A392164 and check the conjectured growth against them.,https://www.erdosproblems.com/1103,
erdos-1104,open,needs-theory,mathematics / graph theory,Let $f(n)$ be the maximum possible chromatic number of a triangle-free graph on $n$ vertices. Estimate $f(n)$.,"Erdős problems database (status: open; prize: no; tags: graph theory, chromatic number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A292528 and check the conjectured growth against them.,https://www.erdosproblems.com/1104,
erdos-1106,open,needs-theory,mathematics / number theory,Let $p(n)$ denote the partition function of $n$ and let $F(n)$ count the number of distinct prime factors of\[\prod_{1\leq k\leq n}p(k).\]Does $F(n)\to \infty$ with $n$? Is $F(n)>n$ for all sufficiently large $n$?,Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A194259, A194260 and check the conjectured growth against them.",https://www.erdosproblems.com/1106,
erdos-1107,open,needs-theory,mathematics / number theory,Let $r\geq 2$. A number $n$ is $r$-powerful if for every prime $p$ which divides $n$ we have $p^r\mid n$. Is every large integer the sum of at most $r+1$ many $r$-powerful numbers?,"Erdős problems database (status: open; prize: no; tags: number theory, powerful)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A056828, A392342, A392343, possible and check the conjectured growth against them.",https://www.erdosproblems.com/1107,
erdos-1108,open,needs-theory,mathematics / number theory,"Let\[A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}\textrm{ finite}\right\}.\]If $k\geq 2$, then does $A$ contain only finitely many $k$th powers? Does it contain only finitely many powerful numbers?","Erdős problems database (status: open; prize: no; tags: number theory, factorials)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A051761, A115645, A025494 and check the conjectured growth against them.",https://www.erdosproblems.com/1108,
erdos-1109,open,needs-theory,mathematics / number theory,"Let $f(N)$ be the size of the largest subset $A\subseteq \{1,\ldots,N\}$ such that every $n\in A+A$ is squarefree. Estimate $f(N)$. In particular, is it true that $f(N)\leq N^{o(1)}$, or even $f(N) \leq (\log N)^{O(1)}$?",Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A392164, A392165 and check the conjectured growth against them.",https://www.erdosproblems.com/1109,
erdos-111,open,needs-theory,mathematics / graph theory,If $G$ is a graph let $h_G(n)$ be defined such that any subgraph of $G$ on $n$ vertices can be made bipartite after deleting at most $h_G(n)$ edges. What is the behaviour of $h_G(n)$? Is it true that $h_G(n)/n\to \infty$ for every graph $G$ with chromatic number $\aleph_1$?,"Erdős problems database (status: open; prize: no; tags: graph theory, chromatic number, set theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/111,
erdos-1110,open,needs-theory,mathematics / number theory,"Let $p>q\geq 2$ be two coprime integers. We call $n$ representable if it is the sum of integers of the form $p^kq^l$, none of which divide each other. If $\{p,q\}\neq \{2,3\}$ then what can be said about the density of non-representable numbers? Are there infinitely many coprime non-representable numbers?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1110,
erdos-1111,open,needs-theory,mathematics / graph theory,"If $G$ is a finite graph and $A,B$ are disjoint sets of vertices then we call $A,B$ anticomplete if there are no edges between $A$ and $B$. If $t,c\geq 1$ then there exists $d\geq 1$ such that if $\chi(G)\geq d$ and $\omega(G)<t$ then there are anticomplete sets $A,B$ with $\chi(A)\geq \chi(B)\geq c$.",Erdős problems database (status: open; prize: no; tags: graph theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1111,
erdos-1112,open,needs-theory,mathematics / additive combinatorics,"(LEAN) No explanation available Let $1\leq d_1<d_2$ and $k\geq 3$. Does there exist an integer $r$ such that if $B=\{b_1<\cdots\}$ is a lacunary sequence of positive integers with $b_{i+1}\geq rb_i$ then there exists a sequence of positive integers $A=\{a_1<\cdots\}$ such that\[d_1\leq a_{i+1}-a_i\leq d_2\]for all $i\geq 1$ and $(kA)\cap B=\emptyset$, where $kA$ is the $k$-fold sumset?",Erdős problems database (status: open (lean); prize: no; tags: additive combinatorics),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1112,
erdos-1113,open,needs-theory,mathematics / number theory,A positive odd integer $m$ such that none of $2^km+1$ are prime for $k\geq 0$ is called a Sierpinski number . We say that a set of primes $P$ is a covering set for $m$ if every $2^km+1$ is divisible by some $p\in P$. Are there Sierpinski numbers with no finite covering set of primes?,"Erdős problems database (status: open; prize: no; tags: number theory, covering systems)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A076336 and check the conjectured growth against them.,https://www.erdosproblems.com/1113,
erdos-1117,open,needs-theory,mathematics / analysis,Let $f(z)$ be an entire function which is not a monomial. Let $\nu(r)$ count the number of $z$ with $\lvert z\rvert=r$ such that $\lvert f(z)\rvert=\max_{\lvert z\rvert=r}\lvert f(z)\rvert$. (This is a finite quantity if $f$ is not a monomial.) Is it possible for\[\limsup \nu(r)=\infty?\]Is it possible for\[\liminf \nu(r)=\infty?\],Erdős problems database (status: open; prize: no; tags: analysis),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1117,
erdos-1119,open,needs-theory,mathematics / analysis,"Independent of the usual axioms of set theory (ZFC). Let $\mathfrak{m}$ be an infinite cardinal with $\aleph_0<\mathfrak{m}<\mathfrak{c}=2^{\aleph_0}$. Let $\{f_\alpha\}$ be a family of entire functions such that, for every $z_0\in \mathbb{C}$, there are at most $\mathfrak{m}$ distinct values of $f_\alpha(z_0)$. Must $\{f_\alpha\}$ have cardinality at most $\mathfrak{m}$?","Erdős problems database (status: independent; prize: no; tags: analysis, set theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1119,
erdos-112,open,needs-theory,mathematics / graph theory,"Let $k=k(n,m)$ be minimal such that any directed graph on $k$ vertices must contain either an independent set of size $n$ or a transitive tournament of size $m$. Determine $k(n,m)$.","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/112,
erdos-1120,open,needs-theory,mathematics / analysis,"Let $f\in \mathbb{C}[z]$ be a monic polynomial of degree $n$, all of whose roots satisfy $\lvert z\rvert\leq 1$. Let\[E= \{ z : \lvert f(z)\rvert \leq 1\}.\]What is the shortest length of a path in $E$ joining $z=0$ to $\lvert z\rvert =1$?",Erdős problems database (status: open; prize: no; tags: analysis),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1120,
erdos-1122,open,needs-theory,mathematics / number theory,"Let $f:\mathbb{N}\to \mathbb{R}$ be an additive function (i.e. $f(ab)=f(a)+f(b)$ whenever $(a,b)=1$). Let\[A=\{ n \geq 1: f(n+1)< f(n)\}.\]If $\lvert A\cap [1,X]\rvert =o(X)$ then must $f(n)=c\log n$ for some $c\in \mathbb{R}$?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1122,
erdos-1123,open,needs-theory,mathematics / algebra,"Independent of the usual axioms of set theory (ZFC). - $100 Let $B_1$ be the Boolean algebra of sets of integers modulo sets of density $0$ (that is, in which two sets are equivalent if and only if they differ by a set of density $0$) and let $B_2$ be the Boolean algebra of sets modulo sets of logarithmic density $0$. Prove that $B_1$ and $B_2$ are not isomorphic.",Erdős problems database (status: independent; prize: $100; tags: algebra),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1123,
erdos-1127,open,needs-theory,mathematics / geometry,"Independent of the usual axioms of set theory (ZFC). Can $\mathbb{R}^n$ be decomposed into countably many sets, such that within each set all the pairwise distances are distinct?","Erdős problems database (status: independent; prize: no; tags: geometry, distances, set theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1127,
erdos-1131,open,needs-theory,mathematics / analysis,"For $x_1,\ldots,x_n\in [-1,1]$ let\[l_k(x)=\frac{\prod_{i\neq k}(x-x_i)}{\prod_{i\neq k}(x_k-x_i)},\]which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$. What is the minimal value of\[I(x_1,\ldots,x_n)=\int_{-1}^1 \sum_k \lvert l_k(x)\rvert^2\mathrm{d}x?\]In particular, is it true that\[\min I =2-(1+o(1))\frac{1}{n}?\]","Erdős problems database (status: open; prize: no; tags: analysis, polynomials)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1131,
erdos-1132,open,needs-theory,mathematics / analysis,"For $x_1,\ldots,x_n\in [-1,1]$ let\[l_k(x)=\frac{\prod_{i\neq k}(x-x_i)}{\prod_{i\neq k}(x_k-x_i)},\]which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$. Let $x_1,x_2,\ldots\in [-1,1]$ be an infinite sequence, and let\[L_n(x) = \sum_{1\leq k\leq n}\lvert l_k(x)\rvert,\]where each $l_k(x)$ is defined above with respect to $x_1,\ldots,x_n$. Must there exist $x\in (-1,1)$ such that\[L_n(x) >\frac{2}{\pi}\log n-O(1)\]for infinitely many $n$? Is it true that\[\limsup_{n\to \infty}\frac{L_n(x)}{\log n}\geq \frac{2}{\pi}\]for almost all $x\in (-1,1)$?","Erdős problems database (status: open; prize: no; tags: analysis, polynomials)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1132,
erdos-1133,open,needs-theory,mathematics / analysis,"Let $C>0$. There exists $\epsilon>0$ such that if $n$ is sufficiently large the following holds. For any $x_1,\ldots,x_n\in [-1,1]$ there exist $y_1,\ldots,y_n\in [-1,1]$ such that, if $P$ is a polynomial of degree $m<(1+\epsilon)n$ with $P(x_i)=y_i$ for at least $(1-\epsilon)n$ many $1\leq i\leq n$, then\[\max_{x\in [-1,1]}\lvert P(x)\rvert >C.\]","Erdős problems database (status: open; prize: no; tags: analysis, polynomials)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1133,
erdos-1135,open,needs-theory,mathematics / number theory,Define $f:\mathbb{N}\to \mathbb{N}$ by $f(n)=n/2$ if $n$ is even and $f(n)=\frac{3n+1}{2}$ if $n$ is odd. Given any integer $m\geq 1$ does there exist $k\geq 1$ such that $f^{(k)}(m)=1$?,Erdős problems database (status: open; prize: $500; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A006370, A008908 and check the conjectured growth against them.",https://www.erdosproblems.com/1135,
erdos-1137,open,needs-theory,mathematics / number theory,"Let $d_n=p_{n+1}-p_n$, where $p_n$ denotes the $n$th prime. Is it true that\[\frac{\max_{n<x}d_{n}d_{n-1}}{(\max_{n<x}d_n)^2}\to 0\]as $x\to \infty$?","Erdős problems database (status: open; prize: no; tags: number theory, primes)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A083550, A005250 and check the conjectured growth against them.",https://www.erdosproblems.com/1137,
erdos-1139,open,needs-theory,mathematics / number theory,Let $1\leq u_1<u_2<\cdots$ be the sequence of integers with at most $2$ prime factors. Is it true that\[\limsup \frac{u_{k+1}-u_k}{\log k}=\infty?\],"Erdős problems database (status: open; prize: no; tags: number theory, primes)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A037143, A101041 and check the conjectured growth against them.",https://www.erdosproblems.com/1139,
erdos-114,open,compute-checkable-small-cases,mathematics / polynomials,"Open, but could be disproved with a finite counterexample. - $250 If $p(z)\in\mathbb{C}[z]$ is a monic polynomial of degree $n$ then is the length of the curve $\{ z\in \mathbb{C} : \lvert p(z)\rvert=1\}$ maximised when $p(z)=z^n-1$?","Erdős problems database (status: falsifiable; prize: $250; tags: polynomials, analysis)",The database marks this as refutable by a finite computation if false: search for a counterexample. Extend OEIS N/A first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/114,
erdos-1142,open,needs-theory,mathematics / number theory,Are there infinitely many $n$ (or any $n>105$) such that $n-2^k$ is prime for all $1<2^k<n$?,"Erdős problems database (status: open; prize: no; tags: number theory, primes)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A039669 and check the conjectured growth against them.,https://www.erdosproblems.com/1142,
erdos-1143,open,needs-theory,mathematics / number theory,"Let $p_1<\cdots<p_u$ be primes and let $k\geq 1$. Let $F_k(p_1,\ldots,p_u)$ be such that every interval of $k$ positive integers contains at least $F_k(p_1,\ldots,p_u)$ multiples of at least one of the $p_i$. Estimate $F_k(p_1,\ldots,p_u)$, particularly in the range $k=\alpha p_u$ for constant $\alpha>2$.","Erdős problems database (status: open; prize: no; tags: number theory, primes)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1143,
erdos-1144,open,needs-theory,mathematics / number theory,"Let $f$ be a random completely multiplicative function, where for each prime $p$ we independently choose $f(p)\in \{-1,1\}$ uniformly at random. Is it true that\[\limsup_{N\to \infty}\frac{\sum_{m\leq N}f(m)}{\sqrt{N}}=\infty\]with probability $1$?","Erdős problems database (status: open; prize: no; tags: number theory, probability)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1144,
erdos-1145,open,needs-theory,mathematics / additive combinatorics,Let $A=\{1\leq a_1<a_2<\cdots\}$ and $B=\{1\leq b_1<b_2<\cdots\}$ be sets of integers with $a_n/b_n\to 1$. If $A+B$ contains all sufficiently large positive integers then is it true that $\limsup 1_A\ast 1_B(n)=\infty$?,"Erdős problems database (status: open; prize: no; tags: additive combinatorics, additive basis)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1145,
erdos-1146,open,needs-theory,mathematics / number theory,"We say that $A\subset \mathbb{N}$ is an essential component if $d_s(A+B)>d_s(B)$ for every $B\subset \mathbb{N}$ with $0<d_s(B)<1$ where $d_s$ is the Schnirelmann density . Is $B=\{2^m3^n : m,n\geq 0\}$ an essential component?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1146,
erdos-1150,open,needs-theory,mathematics / analysis,"Does there exist a constant $c>0$ such that, for all large $n$ and all polynomials $P$ of degree $n$ with coefficients $\pm 1$,\[\max_{\lvert z\rvert=1}\lvert P(z)\rvert > (1+c)\sqrt{n}?\]","Erdős problems database (status: open; prize: no; tags: analysis, polynomials)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1150,
erdos-1151,open,needs-theory,mathematics / analysis,"Given $a_1,\ldots,a_n\in [-1,1]$ let\[\mathcal{L}^nf(x) = \sum_{1\leq i\leq n}f(a_i)\ell_i(x)\]be the unique polynomial of degree $n-1$ which agrees with $f$ on $a_i$ for $1\leq i\leq n$ (that is, the Lagrange interpolation polynomial). Let $a_i$ be the set of Chebyshev nodes . Prove that, for any closed $A\subseteq [-1,1]$, there exists a continuous function $f$ such that $A$ is the set of limit points of $\mathcal{L}^nf(x)$.","Erdős problems database (status: open; prize: no; tags: analysis, polynomials)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1151,
erdos-1152,open,needs-theory,mathematics / analysis,"For $n\geq 1$ fix some sequence of $n$ distinct numbers $x_{1n},\ldots,x_{nn}\in [-1,1]$. Let $\epsilon=\epsilon(n)\to 0$. Does there always exist a continuous function $f:[-1,1]\to \mathbb{R}$ such that if $p_n$ is a sequence of polynomials, with degrees $\deg p_n<(1+\epsilon(n))n$, such that $p_n(x_{kn})=f(x_{kn})$ for all $1\leq k\leq n$, then $p_n(x)\not\to f(x)$ for almost all $x\in [-1,1]$?","Erdős problems database (status: open; prize: no; tags: analysis, polynomials)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1152,
erdos-1154,open,needs-theory,mathematics / analysis,"Open in general, but there exist models of set theory where the result is true. Does there exist, for every $\alpha \in [0,1]$, a ring or field in $\mathbb{R}$ with Hausdorff dimension $\alpha$?",Erdős problems database (status: not disprovable; prize: no; tags: analysis),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1154,
erdos-1155,open,needs-theory,mathematics / graph theory,"Construct a random graph on $n$ vertices in the following way: begin with the complete graph $K_n$. At each stage, choose uniformly a random triangle in the graph and delete all the edges of this triangle. Repeat until the graph is triangle-free. Describe the typical parameters and structure of such a graph. In particular, if $f(n)$ is the number of edges remaining, then is it true that\[\mathbb{E}f(n)\asymp n^{3/2}\]and that $f(n) \ll n^{3/2}$ almost surely?",Erdős problems database (status: open; prize: no; tags: graph theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1155,
erdos-1156,open,needs-theory,mathematics / graph theory,"Let $G$ be a random graph on $n$ vertices, in which every edge is included independently with probability $1/2$. Is there some constant $C$ such that that chromatic number $\chi(G)$ is, almost surely, concentrated on at most $C$ values? Is it true that, if $\omega(n)\to \infty$ sufficiently slowly, then for every function $f(n)$\[\mathbb{P}(\lvert\chi(G)-f(n)\rvert<\omega(n))<1/2\]if $n$ is sufficiently large?","Erdős problems database (status: open; prize: no; tags: graph theory, chromatic number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1156,
erdos-1157,open,needs-theory,mathematics / hypergraphs,"Let $t,k,r\geq 2$. Let $\mathcal{F}$ be the family of all $r$-uniform hypergraphs with $k$ vertices and $s$ edges. Determine\[\mathrm{ex}_r(n,\mathcal{F}).\]","Erdős problems database (status: open; prize: no; tags: hypergraphs, turan number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1157,
erdos-1158,open,needs-theory,mathematics / hypergraphs,"Let $K_{t}(r)$ be the complete $t$-partite $t$-uniform hypergraph with $r$ vertices in each class. Is it true that\[\mathrm{ex}_t(n,K_t(r)) \geq n^{t-r^{1-t}-o(1)}\]for all $t,r$?","Erdős problems database (status: open; prize: no; tags: hypergraphs, turan number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1158,
erdos-1159,open,needs-theory,mathematics / combinatorics,Determine whether there exists a constant $C>1$ such that the following holds. Let $P$ be a finite projective plane . Must there exist a set of points $S$ such that $1\leq \lvert S\cap \ell\rvert \leq C$ for all lines $\ell$?,Erdős problems database (status: open; prize: no; tags: combinatorics),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1159,
erdos-1160,open,needs-theory,mathematics / group theory,Let $g(n)$ denote the number of groups of order $n$. If $n\leq 2^m$ then $g(n)\leq g(2^m)$.,Erdős problems database (status: open; prize: no; tags: group theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A000001 and check the conjectured growth against them.,https://www.erdosproblems.com/1160,
erdos-1162,open,needs-theory,mathematics / group theory,Give an asymptotic formula for the number of subgroups of $S_n$. Is there a statistical theorem on their order?,Erdős problems database (status: open; prize: no; tags: group theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1162,
erdos-1163,open,needs-theory,mathematics / group theory,Describe (by statistical means) the arithmetic structure of the orders of subgroups of $S_n$.,Erdős problems database (status: open; prize: no; tags: group theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1163,
erdos-1167,open,needs-theory,mathematics / set theory,"Let $r\geq 2$ be finite and $\lambda$ be an infinite cardinal. Let $\kappa_\alpha$ be cardinals for all $\alpha<\gamma$. Is it true that\[2^\lambda \to (\kappa_\alpha+1)_{\alpha<\gamma}^{r+1}\]implies\[\lambda \to (\kappa_\alpha)_{\alpha<\gamma}^{r}?\]Here $+$ means cardinal addition, so that $\kappa_\alpha+1=\kappa_\alpha$ if $\kappa_\alpha$ is infinite.","Erdős problems database (status: open; prize: no; tags: set theory, probability)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1167,
erdos-1168,open,needs-theory,mathematics / set theory,"Prove that\[\aleph_{\omega+1}\not\to (\aleph_{\omega+1}, 3,\ldots,3)_{\aleph_0}^2\]without assuming the generalised continuum hypothesis.","Erdős problems database (status: open; prize: no; tags: set theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1168,
erdos-1169,open,needs-theory,mathematics / set theory,"Open in general, but there exist models of set theory where the result is true. Is it true that, for all finite $k<\omega$,\[\omega_1^2 \not\to (\omega_1^2, 3)^2?\]","Erdős problems database (status: not disprovable; prize: no; tags: set theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1169,
erdos-117,open,needs-theory,mathematics / group theory,Let $h(n)$ be minimal such that any group $G$ with the property that any subset of $>n$ elements contains some $x\neq y$ such that $xy=yx$ can be covered by at most $h(n)$ many Abelian subgroups. Estimate $h(n)$ as well as possible.,Erdős problems database (status: open; prize: no; tags: group theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/117,
erdos-1170,open,needs-theory,mathematics / set theory,Is it consistent that\[\omega_2\to (\alpha)_2^2\]for every $\alpha <\omega_2$?,"Erdős problems database (status: open; prize: no; tags: set theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1170,
erdos-1171,open,needs-theory,mathematics / set theory,"Is it true that, for all finite $k<\omega$,\[\omega_1^2\to (\omega_1\omega, 3,\ldots,3)_{k+1}^2?\]","Erdős problems database (status: open; prize: no; tags: set theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1171,
erdos-1172,open,needs-theory,mathematics / set theory,"Establish whether the following are true assuming the generalised continuum hypothesis:\[\omega_3 \to (\omega_2,\omega_1+2)^2,\]\[\omega_3\to (\omega_2+\omega_1,\omega_2+\omega)^2,\]\[\omega_2\to (\omega_1^{\omega+2}+2, \omega_1+2)^2.\]Establish whether the following is consistent with the generalised continuum hypothesis:\[\omega_2\to (\omega_1+\omega)_2^2,\]or even $\omega_2 \to (\xi)_2^2$ for all $\xi<\omega_2$.","Erdős problems database (status: open; prize: no; tags: set theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1172,
erdos-1173,open,needs-theory,mathematics / set theory,Assume the generalised continuum hypothesis. Let\[f: \omega_{\omega+1}\to [\omega_{\omega+1}]^{\leq \aleph_\omega}\]be a set mapping such that\[\lvert f(\alpha)\cap f(\beta)\rvert <\aleph_\omega\]for all $\alpha\neq \beta$. Does there exist a free set of cardinality $\aleph_{\omega+1}$?,"Erdős problems database (status: open; prize: no; tags: set theory, combinatorics)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1173,
erdos-1174,open,needs-theory,mathematics / set theory,"Open in general, but there exist models of set theory where the result is true. Does there exist a graph $G$ with no $K_4$ such that every edge colouring of $G$ with countably many colours contains a monochromatic $K_3$? Does there exist a graph $G$ with no $K_{\aleph_1}$ such that every edge colouring of $G$ with countably many colours contains a monochromatic $K_{\aleph_0}$?","Erdős problems database (status: not disprovable; prize: no; tags: set theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1174,
erdos-1175,open,needs-theory,mathematics / set theory,Let $\kappa$ be an uncountable cardinal. Must there exist a cardinal $\lambda$ such that every graph with chromatic number $\lambda$ contains a triangle-free subgraph with chromatic number $\kappa$?,"Erdős problems database (status: open; prize: no; tags: set theory, chromatic number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1175,
erdos-1176,open,needs-theory,mathematics / set theory,"Open in general, but there exist models of set theory where the result is true. Let $G$ be a graph with chromatic number $\aleph_1$. Is it true that there is a colouring of the edges with $\aleph_1$ many colours such that, in any countable colouring of the vertices, there exists a vertex colour containing all edge colours?","Erdős problems database (status: not disprovable; prize: no; tags: set theory, chromatic number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1176,
erdos-1177,open,needs-theory,mathematics / set theory,"Let $G$ be a finite $3$-uniform hypergraph, and let $F_G(\kappa)$ denote the collection of $3$-uniform hypergraphs with chromatic number $\kappa$ not containing $G$. If $F_G(\aleph_1)$ is not empty then there exists $X\in F_G(\aleph_1)$ of cardinality at most $2^{2^{\aleph_0}}$. If both $F_G(\aleph_1)$ and $F_H(\aleph_1)$ are non-empty then $F_G(\aleph_1)\cap F_H(\aleph_1)$ is non-empty. If $\kappa,\lambda$ are uncountable cardinals and $F_G(\kappa)$ is non-empty then $F_G(\lambda)$ is non-empty.","Erdős problems database (status: open; prize: no; tags: set theory, chromatic number, hypergraphs)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1177,
erdos-1178,open,needs-theory,mathematics / graph theory,"For $r\geq 3$ let $d_r(e)$ be the minimal $d$ such that\[\mathrm{ex}_r(n,\mathcal{F})=o(n^2),\]where $\mathcal{F}$ is the family of $r$-uniform hypergraphs on $d$ vertices with $e$ edges. Prove that\[d_r(e)=(r-2)e+3\]for all $r,e\geq 3$.","Erdős problems database (status: open; prize: no; tags: graph theory, hypergraphs)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/1178,
erdos-1181,open,needs-theory,mathematics / number theory,"Let $q(n,k)$ denote the least prime which does not divide $\prod_{1\leq i\leq k}(n+i)$. Is it true that there exists some $c>0$ such that, for all large $n$,\[q(n,\log n)<(1-c)(\log n)^2?\]",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1181,
erdos-1182,open,needs-theory,mathematics / graph theory,"Let $f(n)$ be maximal such that there is a connected graph $G$ with $n$ vertices and $f(n)$ edges such that\[R(K_3,G)= 2n-1.\]Let $F(n)$ be maximal such that every connected graph $G$ with $n$ vertices and $\leq F(n)$ edges has\[R(K_3,G)= 2n-1.\]Estimate $f(n)$ and $F(n)$. In particular, is it true that $F(n)/n\to \infty$?","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1182,
erdos-1183,open,needs-theory,mathematics / combinatorics,"Let $f(n)$ be maximal such that in any $2$-colouring of the subsets of $\{1,\ldots,n\}$ there is always a monochromatic family of at least $f(n)$ sets which is closed under taking unions and intersections. Estimate $f(n)$. Let $F(n)$ be defined similarly, except that we only require the family be closed under taking unions. Estimate $F(n)$. In particular, is it true that $F(n)\geq n^{\omega(n)}$ for some $\omega(n)\to \infty$ as $n\to \infty$, and $F(n)<(1+o(1))^n$?","Erdős problems database (status: open; prize: no; tags: combinatorics, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1183,
erdos-1184,open,needs-theory,mathematics / number theory,"Let $f(n,k)$ count the number of $1\leq i\leq k$ such that $P(n+i)>k$ (where $P(m)$ is the largest prime divisor of $m$). Is it true that, if $\alpha>1$ is such that $n=k^{\alpha+o(1)}$, then\[f(n,k)=(1-\rho(\alpha)+o(1))k,\]where $\rho$ is the Dickman function ?","Erdős problems database (status: open; prize: no; tags: number theory, primes)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1184,
erdos-1186,open,needs-theory,mathematics / additive combinatorics,"Let $\delta_k$ be such that in any $2$-colouring of $\{1,\ldots,n\}$ there exist at least $(\delta_k+o(1))n^2$ many monochromatic $k$-term arithmetic progressions. Give reasonable bounds (or even an asymptotic formula) for $\delta_k$.","Erdős problems database (status: open; prize: no; tags: additive combinatorics, arithmetic progressions)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1186,
erdos-1188,open,needs-theory,mathematics / number theory,"Call a set of distinct integers $1<n_1<\cdots<n_k$ with associated congruence classes $a_i\pmod{n_i}$ a distinct covering system if every integer satisfies at least one of these congruences. A minimal distinct covering system is one such that no proper subset forms a covering system. Let $F(x)$ count the number of minimal distinct covering systems with all moduli in $[1,x]$. Estimate $F(x)$.","Erdős problems database (status: open; prize: no; tags: number theory, covering systems)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1188,
erdos-1189,open,needs-theory,mathematics / number theory,"Call a set of distinct integers $1<n_1<\cdots<n_k$ a covering set if there is a choice of $a_i\pmod{n_i}$ for $1\leq i\leq k$ such that every integer satisfies at least one of these congruences. A set is an irreducible covering set if no proper subset is a covering set. How many irreducible covering sets of size $k$ are there? What is the minimum and maximum that $n_k$ can be? Determine or estimate $\max \sum\frac{1}{n_i}$, where the maximum ranges over all irreducible covering sets of size $k$. Are there infinitely many $n$ such that the divisors of $n$ (which are $>1$) form an irreducible covering set?","Erdős problems database (status: open; prize: no; tags: number theory, covering systems)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1189,
erdos-1191,open,needs-theory,mathematics / additive combinatorics,"Let $A\subset\mathbb{N}$ be an infinite Sidon set. Is it true that\[\liminf_{x\to \infty} \frac{\lvert A\cap [1,x]\rvert}{x^{1/2}}(\log x)^{1/2}=0?\]Does there exist an infinite Sidon set $A$ such that\[\liminf_{x\to \infty} \frac{\lvert A\cap [1,x]\rvert}{x^{1/2}}(\log x)^c>0\]for some $c>0$?","Erdős problems database (status: open; prize: $1000; tags: additive combinatorics, sidon sets)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1191,
erdos-1192,open,needs-theory,mathematics / additive combinatorics,"For $A\subset \mathbb{N}$ let $f_r(n)$ count the number of solutions to $n=a_1+\cdots+a_r$ with $a_i\in A$. Does there exist, for all $r\geq 2$, a basis $A$ of order $r$ (so that $f_r(n)>0$ for all large $n$) such that\[\sum_{n\leq x}f_r(n)^2 \ll x\]for all $x$?","Erdős problems database (status: open; prize: no; tags: additive combinatorics, additive basis)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1192,
erdos-1194,open,needs-theory,mathematics / additive combinatorics,"Let $A\subset\mathbb{N}$ be such that every integer $n\geq 1$ can be written uniquely as $a_n-b_n$ for some $a_n,b_n\in A$. How fast must $a_n/n$ increase?","Erdős problems database (status: open; prize: no; tags: additive combinatorics, additive basis, sidon sets)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1194,
erdos-1199,open,needs-theory,mathematics / additive combinatorics,Is it true that in any $2$-colouring of $\mathbb{N}$ there exists an infinite set $A$ such that all elements of $A+A$ are the same colour?,"Erdős problems database (status: open; prize: no; tags: additive combinatorics, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1199,
erdos-12,open,needs-theory,mathematics / number theory,"Let $A$ be an infinite set such that there are no distinct $a,b,c\in A$ such that $a\mid (b+c)$ and $b,c>a$. Is there such an $A$ with\[\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}>0?\]Does there exist some absolute constant $c>0$ such that there are always infinitely many $N$ with\[\lvert A\cap\{1,\ldots,N\}\rvert<N^{1-c}?\]Is it true that\[\sum_{n\in A}\frac{1}{n}<\infty?\]",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/12,
erdos-120,open,needs-theory,mathematics / combinatorics,"Let $A\subseteq\mathbb{R}$ be an infinite set. Must there be a set $E\subset \mathbb{R}$ of positive measure which does not contain any set of the shape $aA+b$ for some $a,b\in\mathbb{R}$ and $a\neq 0$?",Erdős problems database (status: open; prize: $100; tags: combinatorics),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/120,
erdos-1200,open,needs-theory,mathematics / number theory,There exists a constant $C$ such that for all large $x$ there is a collection of primes $p_1<\ldots<p_k<x$ with $\sum\frac{1}{p_i}<C$ together with a system of congruences $a_i\pmod{p_i}$ such that every integer $n<x$ satisfies at least one of these congruences.,"Erdős problems database (status: open; prize: no; tags: number theory, primes)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1200,
erdos-1201,open,needs-theory,mathematics / number theory,"Is it true that for every $\epsilon,\eta>0$ there exists a $k$ such that the density of $n$ for which\[P(n(n+1)\cdots(n+k))>n^{1-\epsilon}\]is at least $1-\eta$ (where $P(m)$ is the greatest prime divisor of $m$)?","Erdős problems database (status: open; prize: no; tags: number theory, primes)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1201,
erdos-1203,open,needs-theory,mathematics / number theory,If $\omega(n)$ counts the number of distinct prime divisors of $n$ then let\[F(n)=\max_k \omega(n+k)\frac{\log\log k}{\log k}.\]Prove that $F(n)\to \infty$ as $n\to \infty$.,Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1203,
erdos-1204,open,needs-theory,mathematics / number theory,"We call a sequence of integers $0\leq a_1<\cdots <a_k$ admissible if it is missing at least one congruence class modulo every prime $p$. Let $A(k)=\min a_k$. Estimate $A(k)$ - in particular, is it true that\[A(k)\sim k\log k?\]Estimate\[B(k)=\min \frac{a_1+\cdots+a_k}{k}.\]",Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A008407, A023193, A135311, possible and check the conjectured growth against them.",https://www.erdosproblems.com/1204,
erdos-1206,open,needs-theory,mathematics / number theory,"Does $\{1,2^3,\ldots,N^3\}$ contain a Sidon set of size $\gg N$? Is there an infinite set $A\subset \mathbb{N}$ of positive density such that $\{a^3 : a\in A\}$ is a Sidon set?","Erdős problems database (status: open; prize: no; tags: number theory, sidon sets)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1206,
erdos-1207,open,needs-theory,mathematics / geometry,"Let $P_d(n)$ be such that in any set of $n$ points in $\mathbb{R}^d$ there exist at least $P_d(n)$ many points which do not contain an isosceles triangle. Estimate $P_d(n)$ - in particular, is it true that\[P_2(n)<n^{1-c}\]for some constant $c>0$?","Erdős problems database (status: open; prize: no; tags: geometry, distances)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1207,
erdos-1208,open,needs-theory,mathematics / geometry,For $d\geq 2$ let $F_d(n)$ be minimal such that every set of $n$ points in $\mathbb{R}^d$ contains a set of $F_d(n)$ points with distinct distances. Estimate $F_d(n)$ for fixed $d$ as $n\to \infty$.,"Erdős problems database (status: open; prize: no; tags: geometry, distances)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A193838, A271490, possible and check the conjectured growth against them.",https://www.erdosproblems.com/1208,
erdos-1209,open,needs-theory,mathematics / number theory,"Let $A=\{a_1<a_2<\cdots\}$ be a sequence of integers which tends to infinity sufficiently fast. If there is an $n$ such that all $n+a_k$ are primes then must there exist infinitely many such $n$? What if we ask for $n+a_k$ to be squarefree instead of prime? Are there $n$ such that $n+2^{2^k}$ is always a prime (or always squarefree, or infinitely often a prime, or infinitely often squarefree)?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1209,
erdos-1210,open,needs-theory,mathematics / number theory,"Let $A\subseteq [1,n)$ be a set of integers such that $(a,b)=1$ for all distinct $a,b\in A$. Is it true that\[\sum_{a\in A}\frac{1}{n-a}\leq \sum_{p<n}\frac{1}{p}+O(1)?\]",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1210,
erdos-1212,open,needs-theory,mathematics / number theory,"Let $G$ be the graph with vertex set those pairs $(x,y)\in \mathbb{N}^2$ with $\mathrm{gcd}(x,y)=1$, in which we join two vertices if the differ in only one coordinate, and there by $\pm 1$. Is there a path going to infinity on $G$, say $P$, such that for all $(x,y)\in P$ both $\min(x,y)>1$ and at least one of $x$ or $y$ is composite?","Erdős problems database (status: open; prize: no; tags: number theory, primes)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/1212,
erdos-122,open,needs-theory,mathematics / number theory,"For which number theoretic functions $f$ is it true that, for any $F(n)$ such that $F(n)/f(n)\to 0$ for almost all $n$, there are infinitely many $x$ such that\[\frac{\#\{ n\in \mathbb{N} : n+f(n)\in (x,x+F(x))\}}{F(x)}\to \infty?\]",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/122,
erdos-124,open,needs-theory,mathematics / number theory,"For any $d\geq 1$ and $k\geq 0$ let $P(d,k)$ be the set of integers which are the sum of distinct powers $d^i$ with $i\geq k$. Let $3\leq d_1<d_2<\cdots <d_r$ be integers such that\[\sum_{1\leq i\leq r}\frac{1}{d_r-1}\geq 1.\]Can all sufficiently large integers be written as a sum of the shape $\sum_i c_ia_i$ where $c_i\in \{0,1\}$ and $a_i\in P(d_i,0)$? If we further have $\mathrm{gcd}(d_1,\ldots,d_r)=1$ then, for any $k\geq 1$, can all sufficiently large integers be written as a sum of the shape $\sum_i c_ia_i$ where $c_i\in \{0,1\}$ and $a_i\in P(d_i,k)$?","Erdős problems database (status: open; prize: no; tags: number theory, base representations)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/124,
erdos-126,open,needs-theory,mathematics / number theory,Let $f(n)$ be maximal such that if $A\subseteq\mathbb{N}$ has $\lvert A\rvert=n$ then $\prod_{a\neq b\in A}(a+b)$ has at least $f(n)$ distinct prime factors. Is it true that $f(n)/\log n\to\infty$?,Erdős problems database (status: open; prize: $250; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/126,
erdos-128,open,compute-checkable-small-cases,mathematics / graph theory,"Open, but could be disproved with a finite counterexample. - $250 Let $G$ be a graph with $n$ vertices such that every induced subgraph on $\geq \lfloor n/2\rfloor$ vertices has more than $n^2/50$ edges. Must $G$ contain a triangle?",Erdős problems database (status: falsifiable; prize: $250; tags: graph theory),The database marks this as refutable by a finite computation if false: search for a counterexample. Extend OEIS N/A first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/128,
erdos-129,open,needs-theory,mathematics / graph theory,"Let $R(n;k,r)$ be the smallest $N$ such that if the edges of $K_N$ are $r$-coloured then there is a set of $n$ vertices which does not contain a copy of $K_k$ in at least one of the $r$ colours. Prove that there is a constant $C=C(r)>1$ such that\[R(n;3,r) < C^{\sqrt{n}}.\]","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/129,
erdos-130,open,needs-theory,mathematics / graph theory,"Let $A\subset\mathbb{R}^2$ be an infinite set which contains no three points on a line and no four points on a circle. Consider the graph with vertices the points in $A$, where two vertices are joined by an edge if and only if they are an integer distance apart. How large can the chromatic number and clique number of this graph be? In particular, can the chromatic number be infinite?","Erdős problems database (status: open; prize: no; tags: graph theory, chromatic number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/130,
erdos-131,open,needs-theory,mathematics / number theory,"Let $F(N)$ be the maximal size of $A\subseteq\{1,\ldots,N\}$ such that no $a\in A$ divides the sum of any distinct elements of $A\backslash\{a\}$. Estimate $F(N)$. In particular, is it true that\[F(N) > N^{1/2-o(1)}?\]",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A068063 and check the conjectured growth against them.,https://www.erdosproblems.com/131,
erdos-132,open,needs-theory,mathematics / distances,Let $A\subset \mathbb{R}^2$ be a set of $n$ points. Must there be two distances which occur at least once but between at most $n$ pairs of points? Must the number of such distances $\to \infty$ as $n\to \infty$?,Erdős problems database (status: open; prize: $100; tags: distances),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/132,
erdos-137,open,needs-theory,mathematics / number theory,We say that $N$ is powerful if whenever $p\mid N$ we also have $p^2\mid N$. Let $k\geq 3$. Can the product of any $k$ consecutive positive integers ever be powerful?,Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/137,
erdos-138,open,needs-theory,mathematics / additive combinatorics,"Let the van der Waerden number $W(k)$ be such that whenever $N\geq W(k)$ and $\{1,\ldots,N\}$ is $2$-coloured there must exist a monochromatic $k$-term arithmetic progression. Improve the bounds for $W(k)$ - for example, prove that $W(k)^{1/k}\to \infty$.",Erdős problems database (status: open; prize: $500; tags: additive combinatorics),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A005346 and check the conjectured growth against them.,https://www.erdosproblems.com/138,
erdos-14,open,needs-theory,mathematics / number theory,"Let $A\subseteq \mathbb{N}$. Let $B\subseteq \mathbb{N}$ be the set of integers which are representable in exactly one way as the sum of two elements from $A$. Is it true that for all $\epsilon>0$ and large $N$\[\lvert \{1,\ldots,N\}\backslash B\rvert \gg_\epsilon N^{1/2-\epsilon}?\]Is it possible that\[\lvert \{1,\ldots,N\}\backslash B\rvert =o(N^{1/2})?\]","Erdős problems database (status: open; prize: no; tags: number theory, sidon sets, additive combinatorics)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A143824, possible and check the conjectured growth against them.",https://www.erdosproblems.com/14,
erdos-141,open,needs-theory,mathematics / additive combinatorics,Let $k\geq 3$. Are there $k$ consecutive primes in arithmetic progression?,"Erdős problems database (status: open; prize: no; tags: additive combinatorics, primes, arithmetic progressions)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A006560 and check the conjectured growth against them.,https://www.erdosproblems.com/141,
erdos-142,open,needs-theory,mathematics / additive combinatorics,"Let $r_k(N)$ be the largest possible size of a subset of $\{1,\ldots,N\}$ that does not contain any non-trivial $k$-term arithmetic progression. Prove an asymptotic formula for $r_k(N)$.","Erdős problems database (status: open; prize: $10000; tags: additive combinatorics, arithmetic progressions)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A003002, A003003, A003004, A003005 and check the conjectured growth against them.",https://www.erdosproblems.com/142,
erdos-143,open,needs-theory,mathematics / primitive sets,"Let $A\subset (1,\infty)$ be a countably infinite set such that for all $x\neq y\in A$ and integers $k\geq 1$ we have\[ \lvert kx -y\rvert \geq 1.\]Does this imply that $A$ is sparse? In particular, does this imply that\[\sum_{x\in A}\frac{1}{x\log x}<\infty\]or\[\sum_{\substack{x <n\\ x\in A}}\frac{1}{x}=o(\log n)?\]",Erdős problems database (status: open; prize: $500; tags: primitive sets),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/143,
erdos-145,open,needs-theory,mathematics / number theory,"Let $s_1<s_2<\cdots$ be the sequence of squarefree numbers. Is it true that, for any $\alpha \geq 0$,\[\lim_{x\to \infty}\frac{1}{x}\sum_{s_n\leq x}(s_{n+1}-s_n)^\alpha\]exists?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A005117 and check the conjectured growth against them.,https://www.erdosproblems.com/145,
erdos-148,open,needs-theory,mathematics / number theory,"Let $F(k)$ be the number of solutions to\[ 1= \frac{1}{n_1}+\cdots+\frac{1}{n_k},\]where $1\leq n_1<\cdots<n_k$ are distinct integers. Find good estimates for $F(k)$.","Erdős problems database (status: open; prize: no; tags: number theory, unit fractions)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A076393, A006585 and check the conjectured growth against them.",https://www.erdosproblems.com/148,
erdos-149,open,needs-theory,mathematics / graph theory,"The strong chromatic index of a graph $G$, denoted by $\mathrm{sq}(G)$, is the minimum $k$ such that the edges of $G$ can be partitioned into $k$ sets of 'strongly independent' edges, that is, such that the subgraph of $G$ induced by each set is the union of vertex-disjoint edges. Is it true that, for any graph $G$ with maximum degree $\Delta$,\[\mathrm{sq}(G)\leq\frac{5}{4}\Delta^2?\]",Erdős problems database (status: open; prize: no; tags: graph theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/149,
erdos-15,open,needs-theory,mathematics / number theory,"Is it true that\[\sum_{n=1}^\infty(-1)^n\frac{n}{p_n}\]converges, where $p_n$ is the sequence of primes?","Erdős problems database (status: open; prize: no; tags: number theory, primes)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/15,
erdos-151,open,needs-theory,mathematics / graph theory,For a graph $G$ let $\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ on at least two vertices (sometimes called the clique transversal number). Let $H(n)$ be maximal such that every triangle-free graph on $n$ vertices contains an independent set on $H(n)$ vertices. If $G$ is a graph on $n$ vertices then is\[\tau(G)\leq n-H(n)?\],Erdős problems database (status: open; prize: no; tags: graph theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/151,
erdos-153,open,needs-theory,mathematics / sidon sets,Let $A$ be a finite Sidon set and $A+A=\{s_1<\cdots<s_t\}$. Is it true that\[\frac{1}{t}\sum_{1\leq i<t}(s_{i+1}-s_i)^2 \to \infty\]as $\lvert A\rvert\to \infty$?,Erdős problems database (status: open; prize: no; tags: sidon sets),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/153,
erdos-155,open,needs-theory,mathematics / additive combinatorics,"Let $F(N)$ be the size of the largest Sidon subset of $\{1,\ldots,N\}$. Is it true that for every $k\geq 1$ we have\[F(N+k)\leq F(N)+1\]for all sufficiently large $N$?","Erdős problems database (status: open; prize: no; tags: additive combinatorics, sidon sets)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A143824, A227590, A003022 and check the conjectured growth against them.",https://www.erdosproblems.com/155,
erdos-156,open,needs-theory,mathematics / sidon sets,"Does there exist a maximal Sidon set $A\subset \{1,\ldots,N\}$ of size $O(N^{1/3})$?",Erdős problems database (status: open; prize: no; tags: sidon sets),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A382397 and check the conjectured growth against them.,https://www.erdosproblems.com/156,
erdos-158,open,needs-theory,mathematics / sidon sets,"Let $A\subset \mathbb{N}$ be an infinite set such that, for any $n$, there are most $2$ solutions to $a+b=n$ with $a\leq b$. Must\[\liminf_{N\to\infty}\frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}}=0?\]",Erdős problems database (status: open; prize: no; tags: sidon sets),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/158,
erdos-159,open,needs-theory,mathematics / graph theory,"There exists some constant $c>0$ such that $$R(C_4,K_n) \ll n^{2-c}.$$","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/159,
erdos-160,open,needs-theory,mathematics / additive combinatorics,"Let $h(N)$ be the smallest $k$ such that $\{1,\ldots,N\}$ can be coloured with $k$ colours so that every four-term arithmetic progression must contain at least three distinct colours. Estimate $h(N)$.","Erdős problems database (status: open; prize: no; tags: additive combinatorics, arithmetic progressions)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/160,
erdos-161,open,needs-theory,mathematics / combinatorics,"Let $\alpha\in[0,1/2)$ and $n,t\geq 1$. Let $F^{(t)}(n,\alpha)$ be the smallest $m$ such that we can $2$-colour the edges of the complete $t$-uniform hypergraph on $n$ vertices such that if $X\subseteq [n]$ with $\lvert X\rvert \geq m$ then there are at least $\alpha \binom{\lvert X\rvert}{t}$ many $t$-subsets of $X$ of each colour. For fixed $n,t$ as we change $\alpha$ from $0$ to $1/2$ does $F^{(t)}(n,\alpha)$ increase continuously or are there jumps? Only one jump?","Erdős problems database (status: open; prize: $500; tags: combinatorics, ramsey theory, discrepancy)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/161,
erdos-162,open,needs-theory,mathematics / combinatorics,"Let $\alpha>0$ and $n\geq 1$. Let $F(n,\alpha)$ be the largest $k$ such that there exists some 2-colouring of the edges of $K_n$ in which any induced subgraph $H$ on at least $k$ vertices contains more than $\alpha\binom{\lvert H\rvert}{2}$ many edges of each colour. Prove that for every fixed $0\leq \alpha \leq 1/2$, as $n\to\infty$,\[F(n,\alpha)\sim c_\alpha \log n\]for some constant $c_\alpha$.","Erdős problems database (status: open; prize: no; tags: combinatorics, ramsey theory, discrepancy)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/162,
erdos-165,open,needs-theory,mathematics / graph theory,"Give an asymptotic formula for $R(3,k)$.","Erdős problems database (status: open; prize: $250; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A000791 and check the conjectured growth against them.,https://www.erdosproblems.com/165,
erdos-167,open,compute-checkable-small-cases,mathematics / graph theory,"Open, but could be disproved with a finite counterexample. If $G$ is a graph with at most $k$ edge disjoint triangles then can $G$ be made triangle-free after removing at most $2k$ edges?",Erdős problems database (status: falsifiable; prize: no; tags: graph theory),The database marks this as refutable by a finite computation if false: search for a counterexample. Extend OEIS N/A first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/167,
erdos-168,open,needs-theory,mathematics / additive combinatorics,"Let $F(N)$ be the size of the largest subset of $\{1,\ldots,N\}$ which does not contain any set of the form $\{n,2n,3n\}$. What is\[ \lim_{N\to \infty}\frac{F(N)}{N}?\]Is this limit irrational?",Erdős problems database (status: open; prize: no; tags: additive combinatorics),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A004059, A057561, A094708, A386439 and check the conjectured growth against them.",https://www.erdosproblems.com/168,
erdos-169,open,needs-theory,mathematics / additive combinatorics,Let $k\geq 3$ and $f(k)$ be the supremum of $\sum_{n\in A}\frac{1}{n}$ as $A$ ranges over all sets of positive integers which do not contain a $k$-term arithmetic progression. Estimate $f(k)$. Is\[\lim_{k\to \infty}\frac{f(k)}{\log W(k)}=\infty\]where $W(k)$ is the van der Waerden number?,"Erdős problems database (status: open; prize: no; tags: additive combinatorics, arithmetic progressions)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A005346 and check the conjectured growth against them.,https://www.erdosproblems.com/169,
erdos-17,open,needs-theory,mathematics / number theory,"Are there infinitely many primes $p$ such that every even number $n\leq p-3$ can be written as a difference of primes $n=q_1-q_2$ where $q_1,q_2\leq p$?","Erdős problems database (status: open; prize: no; tags: number theory, primes)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A038133 and check the conjectured growth against them.,https://www.erdosproblems.com/17,
erdos-170,open,needs-theory,mathematics / additive combinatorics,"Let $F(N)$ be the smallest possible size of $A\subset \{0,1,\ldots,N\}$ such that $\{0,1,\ldots,N\}\subset A-A$. Find the value of\[\lim_{N\to \infty}\frac{F(N)}{N^{1/2}}.\]",Erdős problems database (status: open; prize: no; tags: additive combinatorics),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A046693 and check the conjectured growth against them.,https://www.erdosproblems.com/170,
erdos-172,open,needs-theory,mathematics / additive combinatorics,Is it true that in any finite colouring of $\mathbb{N}$ there exist arbitrarily large finite $A$ such that all sums and products of distinct elements in $A$ are the same colour?,"Erdős problems database (status: open; prize: no; tags: additive combinatorics, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/172,
erdos-173,open,needs-theory,mathematics / geometry,"In any $2$-colouring of $\mathbb{R}^2$, for all but at most one triangle $T$, there is a monochromatic congruent copy of $T$.","Erdős problems database (status: open; prize: no; tags: geometry, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/173,
erdos-174,open,needs-theory,mathematics / geometry,"A finite set $A\subset \mathbb{R}^n$ is called Ramsey if, for any $k\geq 1$, there exists some $d=d(A,k)$ such that in any $k$-colouring of $\mathbb{R}^d$ there exists a monochromatic copy of $A$. Characterise the Ramsey sets in $\mathbb{R}^n$.","Erdős problems database (status: open; prize: no; tags: geometry, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/174,
erdos-176,open,needs-theory,mathematics / additive combinatorics,"Let $N(k,\ell)$ be the minimal $N$ such that for any $f:\{1,\ldots,N\}\to\{-1,1\}$ there must exist a $k$-term arithmetic progression $P$ such that\[ \left\lvert \sum_{n\in P}f(n)\right\rvert\geq \ell.\]Find good upper bounds for $N(k,\ell)$. Is it true that for any $c>0$ there exists some $C>1$ such that\[N(k,ck)\leq C^k?\]What about\[N(k,2)\leq C^k\]or\[N(k,\sqrt{k})\leq C^k?\]","Erdős problems database (status: open; prize: no; tags: additive combinatorics, arithmetic progressions, discrepancy)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/176,
erdos-177,open,needs-theory,mathematics / discrepancy,"Find the smallest $h(d)$ such that the following holds. There exists a function $f:\mathbb{N}\to\{-1,1\}$ such that, for every $d\geq 1$,\[\max_{P_d}\left\lvert \sum_{n\in P_d}f(n)\right\rvert\leq h(d),\]where $P_d$ ranges over all finite arithmetic progressions with common difference $d$.","Erdős problems database (status: open; prize: no; tags: discrepancy, arithmetic progressions)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/177,
erdos-18,open,needs-theory,mathematics / number theory,We call $m$ practical if every integer $1\leq n<m$ is the sum of distinct divisors of $m$. If $m$ is practical then let $h(m)$ be such that $h(m)$ many divisors always suffice. Are there infinitely many practical $m$ such that\[h(m) < (\log\log m)^{O(1)}?\]Is it true that $h(n!)<n^{o(1)}$? Or perhaps even $h(n!)<(\log n)^{O(1)}$?,"Erdős problems database (status: open; prize: no; tags: number theory, divisors, factorials)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A005153 and check the conjectured growth against them.,https://www.erdosproblems.com/18,
erdos-181,open,needs-theory,mathematics / graph theory,Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Prove that\[R(Q_n) \ll 2^n.\],"Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/181,
erdos-184,open,needs-theory,mathematics / graph theory,Any graph on $n$ vertices can be decomposed into $O(n)$ many edge-disjoint cycles and edges.,"Erdős problems database (status: open; prize: no; tags: graph theory, cycles)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/184,
erdos-187,open,needs-theory,mathematics / additive combinatorics,"Find the best function $f(d)$ such that, in any 2-colouring of the integers, at least one colour class contains an arithmetic progression with common difference $d$ of length $f(d)$ for infinitely many $d$.","Erdős problems database (status: open; prize: no; tags: additive combinatorics, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/187,
erdos-188,open,needs-theory,mathematics / geometry,"What is the smallest $k$ such that $\mathbb{R}^2$ can be red/blue coloured with no pair of red points unit distance apart, and no $k$-term arithmetic progression of blue points with distance $1$?","Erdős problems database (status: open; prize: no; tags: geometry, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/188,
erdos-19,open,compute-checkable-small-cases,mathematics / graph theory,Resolved up to a finite check. - $500 If $G$ is an edge-disjoint union of $n$ copies of $K_n$ then is $\chi(G)=n$?,"Erdős problems database (status: decidable; prize: $500; tags: graph theory, chromatic number)",The database marks this as reduced to a finite computation: run it. Extend OEIS N/A first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/19,
erdos-193,open,needs-theory,mathematics / geometry,"Let $S\subseteq \mathbb{Z}^3$ be a finite set and let $A=\{a_1,a_2,\ldots,\}\subset \mathbb{Z}^3$ be an infinite $S$-walk, so that $a_{i+1}-a_i\in S$ for all $i$. Must $A$ contain three collinear points?",Erdős problems database (status: open; prize: no; tags: geometry),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A231255 and check the conjectured growth against them.,https://www.erdosproblems.com/193,
erdos-195,open,needs-theory,mathematics / arithmetic progressions,What is the largest $k$ such that in any permutation of $\mathbb{Z}$ there must exist a monotone $k$-term arithmetic progression $x_1<\cdots<x_k$?,Erdős problems database (status: open; prize: no; tags: arithmetic progressions),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/195,
erdos-196,open,needs-theory,mathematics / arithmetic progressions,"Must every permutation of $\mathbb{N}$ contain a monotone 4-term arithmetic progression? In other words, given a permutation $x$ of $\mathbb{N}$ must there be indices with either $i<j<k<l$ or $i>j>k>l$ such that $x_i,x_j,x_k,x_l$ are an arithmetic progression?",Erdős problems database (status: open; prize: no; tags: arithmetic progressions),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/196,
erdos-197,open,needs-theory,mathematics / arithmetic progressions,"Can $\mathbb{N}$ be partitioned into two sets, each of which can be permuted to avoid monotone 3-term arithmetic progressions?",Erdős problems database (status: open; prize: no; tags: arithmetic progressions),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/197,
erdos-20,open,needs-theory,mathematics / combinatorics,"Let $f(n,k)$ be minimal such that every family $\mathcal{F}$ of $n$-uniform sets with $\lvert \mathcal{F}\rvert \geq f(n,k)$ contains a $k$-sunflower. Is it true that\[f(n,k) < c_k^n\]for some constant $c_k>0$?",Erdős problems database (status: open; prize: $1000; tags: combinatorics),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A332077 and check the conjectured growth against them.,https://www.erdosproblems.com/20,
erdos-200,open,needs-theory,mathematics / primes,"Does the longest arithmetic progression of primes in $\{1,\ldots,N\}$ have length $o(\log N)$?","Erdős problems database (status: open; prize: no; tags: primes, arithmetic progressions)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A005115 and check the conjectured growth against them.,https://www.erdosproblems.com/200,
erdos-201,open,needs-theory,mathematics / additive combinatorics,"Let $G_k(N)$ be such that any set of $N$ integers contains a subset of size at least $G_k(N)$ which does not contain a $k$-term arithmetic progression. Determine the size of $G_k(N)$. How does it relate to $R_k(N)$, the size of the largest subset of $\{1,\ldots,N\}$ without a $k$-term arithmetic progression? Is it true that\[\lim_{N\to \infty}\frac{R_3(N)}{G_3(N)}=1?\]","Erdős problems database (status: open; prize: no; tags: additive combinatorics, arithmetic progressions)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A003002, A003003, A003004, A003005, possible and check the conjectured growth against them.",https://www.erdosproblems.com/201,
erdos-203,open,needs-theory,mathematics / primes,"Is there an integer $m\geq 1$ with $(m,6)=1$ such that none of $2^k3^\ell m+1$ are prime, for any $k,\ell\geq 0$?","Erdős problems database (status: open; prize: no; tags: primes, covering systems)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/203,
erdos-208,open,needs-theory,mathematics / number theory,"Let $s_1<s_2<\cdots$ be the sequence of squarefree numbers. Is it true that, for any $\epsilon>0$ and large $n$,\[s_{n+1}-s_n \ll_\epsilon s_n^{\epsilon}?\]Is it true that\[s_{n+1}-s_n \leq (1+o(1))\frac{\pi^2}{6}\frac{\log s_n}{\log\log s_n}?\]",Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A005117, A076259 and check the conjectured growth against them.",https://www.erdosproblems.com/208,
erdos-212,open,needs-theory,mathematics / geometry,Is there a dense subset of $\mathbb{R}^2$ such that all pairwise distances are rational?,"Erdős problems database (status: open; prize: no; tags: geometry, distances)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/212,
erdos-213,open,needs-theory,mathematics / geometry,"Let $n\geq 4$. Are there $n$ points in $\mathbb{R}^2$, no three on a line and no four on a circle, such that all pairwise distances are integers?","Erdős problems database (status: open; prize: no; tags: geometry, distances)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/213,
erdos-217,open,needs-theory,mathematics / geometry,"For which $n$ are there $n$ points in $\mathbb{R}^2$, no three on a line and no four on a circle, which determine $n-1$ distinct distances and so that (in some ordering of the distances) the $i$th distance occurs $i$ times?","Erdős problems database (status: open; prize: no; tags: geometry, distances)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/217,
erdos-218,open,needs-theory,mathematics / number theory,"Let $d_n=p_{n+1}-p_n$. The set of $n$ such that $d_{n+1}\geq d_n$ has density $1/2$, and similarly for $d_{n+1}\leq d_n$. Furthermore, there are infinitely many $n$ such that $d_{n+1}=d_n$.","Erdős problems database (status: open; prize: no; tags: number theory, primes)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A333230, A333231, A064113 and check the conjectured growth against them.",https://www.erdosproblems.com/218,
erdos-222,open,needs-theory,mathematics / number theory,Let $n_1<n_2<\cdots$ be the sequence of integers which are the sum of two squares. Explore the behaviour of (i.e. find good upper and lower bounds for) the consecutive differences $n_{k+1}-n_k$.,"Erdős problems database (status: open; prize: no; tags: number theory, squares)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A001481, A256435 and check the conjectured growth against them.",https://www.erdosproblems.com/222,
erdos-23,open,compute-checkable-small-cases,mathematics / graph theory,"Open, but could be disproved with a finite counterexample. Can every triangle-free graph on $5n$ vertices be made bipartite by deleting at most $n^2$ edges?",Erdős problems database (status: falsifiable; prize: no; tags: graph theory),The database marks this as refutable by a finite computation if false: search for a counterexample. Extend OEIS A389646 first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/23,
erdos-233,open,needs-theory,mathematics / number theory,"Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Prove that\[\sum_{1\leq n\leq N}d_n^2 \ll N(\log N)^2.\]","Erdős problems database (status: open; prize: no; tags: number theory, primes)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A074741 and check the conjectured growth against them.,https://www.erdosproblems.com/233,
erdos-234,open,needs-theory,mathematics / number theory,For every $c\geq 0$ the density $f(c)$ of integers for which\[\frac{p_{n+1}-p_n}{\log n}< c\]exists and is a continuous function of $c$.,"Erdős problems database (status: open; prize: no; tags: number theory, primes)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/234,
erdos-236,open,needs-theory,mathematics / number theory,Let $f(n)$ count the number of solutions to $n=p+2^k$ for prime $p$ and $k\geq 0$. Is it true that $f(n)=o(\log n)$?,"Erdős problems database (status: open; prize: no; tags: number theory, primes)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A039669, A109925 and check the conjectured growth against them.",https://www.erdosproblems.com/236,
erdos-238,open,needs-theory,mathematics / number theory,"Let $c_1,c_2>0$. Is it true that, for any sufficiently large $x$, there exist more than $c_1\log x$ many consecutive primes $\leq x$ such that the difference between any two is $>c_2$?","Erdős problems database (status: open; prize: no; tags: number theory, primes)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/238,
erdos-241,open,needs-theory,mathematics / additive combinatorics,"Let $f(N)$ be the maximum size of $A\subseteq \{1,\ldots,N\}$ such that the sums $a+b+c$ with $a,b,c\in A$ are all distinct (aside from the trivial coincidences). Is it true that\[ f(N)\sim N^{1/3}?\]","Erdős problems database (status: open; prize: $100; tags: additive combinatorics, sidon sets)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A387704 and check the conjectured growth against them.,https://www.erdosproblems.com/241,
erdos-242,open,compute-checkable-small-cases,mathematics / number theory,"Open, but could be disproved with a finite counterexample. For every $n>2$ there exist distinct integers $1\leq x<y<z$ such that\[\frac{4}{n} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z}.\]","Erdős problems database (status: falsifiable; prize: no; tags: number theory, unit fractions)","The database marks this as refutable by a finite computation if false: search for a counterexample. Extend OEIS A073101, A075245, A075246, A075247, A075248, A287116 first; a new term is a checkable contribution even without a resolution.",https://www.erdosproblems.com/242,
erdos-243,open,needs-theory,mathematics / number theory,"Let $1\leq a_1<a_2<\cdots$ be a sequence of integers such that\[\lim_{n\to \infty}\frac{a_n}{a_{n-1}^2}=1\]and $\sum\frac{1}{a_n}\in \mathbb{Q}$. Then, for all sufficiently large $n\geq 1$,\[ a_n = a_{n-1}^2-a_{n-1}+1.\]",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A000058 and check the conjectured growth against them.,https://www.erdosproblems.com/243,
erdos-244,open,needs-theory,mathematics / number theory,"Let $C>1$. Does the set of integers of the form $p+\lfloor C^k\rfloor$, for some prime $p$ and $k\geq 0$, have density $>0$?","Erdős problems database (status: open; prize: no; tags: number theory, primes)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/244,
erdos-247,open,needs-theory,mathematics / number theory,Let $1\leq a_1<a_2<\cdots$ be a sequence of integers such that\[\limsup \frac{a_n}{n}=\infty.\]Is\[\sum_{n=1}^\infty \frac{1}{2^{a_n}}\]transcendental?,"Erdős problems database (status: open; prize: no; tags: number theory, irrationality)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/247,
erdos-249,open,needs-theory,mathematics / number theory,Is\[\sum_n \frac{\phi(n)}{2^n}\]irrational? Here $\phi$ is the Euler totient function .,"Erdős problems database (status: open; prize: no; tags: number theory, irrationality)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A256936 and check the conjectured growth against them.,https://www.erdosproblems.com/249,
erdos-25,open,needs-theory,mathematics / number theory,"Let $1\leq n_1<n_2<\cdots$ be an arbitrary sequence of integers, each with an associated residue class $a_i\pmod{n_i}$. Let $A$ be the set of integers $n$ such that for every $i$ either $n<n_i$ or $n\not\equiv a_i\pmod{n_i}$. Must the logarithmic density of $A$ exist?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/25,
erdos-251,open,needs-theory,mathematics / number theory,Is\[\sum \frac{p_n}{2^n}\]irrational? (Here $p_n$ is the $n$th prime.),"Erdős problems database (status: open; prize: no; tags: number theory, irrationality)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A098990 and check the conjectured growth against them.,https://www.erdosproblems.com/251,
erdos-252,open,needs-theory,mathematics / number theory,Let $k\geq 1$ and $\sigma_k(n)=\sum_{d\mid n}d^k$. Is\[\sum \frac{\sigma_k(n)}{n!}\]irrational?,"Erdős problems database (status: open; prize: no; tags: number theory, irrationality)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A227988, A227989, A307036, A359060, possible and check the conjectured growth against them.",https://www.erdosproblems.com/252,
erdos-254,open,needs-theory,mathematics / number theory,"Let $A\subseteq \mathbb{N}$ be such that\[\lvert A\cap [1,2x]\rvert -\lvert A\cap [1,x]\rvert \to \infty\textrm{ as }x\to \infty\]and\[\sum_{n\in A} \{ \theta n\}=\infty\]for every $\theta\in (0,1)$, where $\{x\}$ is the distance of $x$ from the nearest integer. Then every sufficiently large integer is the sum of distinct elements of $A$.",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/254,
erdos-256,open,needs-theory,mathematics / analysis,"Let $n\geq 1$ and $f(n)$ be maximal such that for any integers $1\leq a_1\leq \cdots \leq a_n$ we have\[\max_{\lvert z\rvert=1}\left\lvert \prod_{i}(1-z^{a_i})\right\rvert\geq f(n).\]Estimate $f(n)$ - in particular, is it true that there exists some constant $c>0$ such that\[\log f(n) \gg n^c?\]",Erdős problems database (status: open; prize: no; tags: analysis),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/256,
erdos-257,open,needs-theory,mathematics / irrationality,Let $A\subseteq \mathbb{N}$ be an infinite set. Is\[\sum_{n\in A}\frac{1}{2^n-1}\]irrational?,Erdős problems database (status: open; prize: no; tags: irrationality),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/257,
erdos-260,open,needs-theory,mathematics / irrationality,Let $a_1<a_2<\cdots$ be an increasing sequence such that $a_n/n\to \infty$. Is the sum\[\sum_n \frac{a_n}{2^{a_n}}\]irrational?,Erdős problems database (status: open; prize: no; tags: irrationality),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/260,
erdos-261,open,needs-theory,mathematics / number theory,"Are there infinitely many $n$ such that there exists some $t\geq 2$ and distinct integers $a_1,\ldots,a_t\geq 1$ such that\[\frac{n}{2^n}=\sum_{1\leq k\leq t}\frac{a_k}{2^{a_k}}?\]Is this true for all $n$? Is there a rational $x$ such that\[x = \sum_{k=1}^\infty \frac{a_k}{2^{a_k}}\]has at least $2^{\aleph_0}$ solutions?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/261,
erdos-263,open,needs-theory,mathematics / irrationality,Let $a_n$ be an increasing sequence of positive integers such that for every sequence of positive integers $b_n$ with $b_n/a_n\to 1$ the sum\[\sum\frac{1}{b_n}\]is irrational. Is $a_n=2^{2^n}$ such a sequence? Must such a sequence satisfy $a_n^{1/n}\to \infty$?,Erdős problems database (status: open; prize: no; tags: irrationality),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/263,
erdos-264,open,needs-theory,mathematics / irrationality,Let $a_n$ be a sequence of positive integers such that for every bounded sequence of integers $b_n$ (with $a_n+b_n\neq 0$ and $b_n\neq 0$ for all $n$) the sum\[\sum \frac{1}{a_n+b_n}\]is irrational. Are $a_n=2^n$ or $a_n=n!$ examples of such a sequence?,Erdős problems database (status: open; prize: no; tags: irrationality),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/264,
erdos-265,open,needs-theory,mathematics / irrationality,Let $1\leq a_1<a_2<\cdots$ be an increasing sequence of integers. How fast can $a_n\to \infty$ grow if\[\sum\frac{1}{a_n}\quad\textrm{and}\quad\sum\frac{1}{a_n-1}\]are both rational?,Erdős problems database (status: open; prize: no; tags: irrationality),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/265,
erdos-267,open,needs-theory,mathematics / irrationality,Let $F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$ be the Fibonacci sequence. Let $n_1<n_2<\cdots $ be an infinite sequence with $n_{k+1}/n_k \geq c>1$. Must\[\sum_k\frac{1}{F_{n_k}}\]be irrational?,Erdős problems database (status: open; prize: no; tags: irrationality),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/267,
erdos-269,open,needs-theory,mathematics / irrationality,"Let $P$ be a finite set of primes with $\lvert P\rvert \geq 2$ and let $\{a_1<a_2<\cdots\}=\{ n\in \mathbb{N} : \textrm{if }p\mid n\textrm{ then }p\in P\}$. Is the sum\[\sum_{n=1}^\infty \frac{1}{[a_1,\ldots,a_n]},\]where $[a_1,\ldots,a_n]$ is the lowest common multiple of $a_1,\ldots,a_n$, irrational?",Erdős problems database (status: open; prize: no; tags: irrationality),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/269,
erdos-271,open,needs-theory,mathematics / additive combinatorics,"Let $A(n)=\{a_0<a_1<\cdots\}$ be the sequence defined by $a_0=0$ and $a_1=n$, and for $k\geq 1$ define $a_{k+1}$ as the least positive integer such that there is no three-term arithmetic progression in $\{a_0,\ldots,a_{k+1}\}$. Can the $a_k$ be explicitly determined? How fast do they grow?","Erdős problems database (status: open; prize: no; tags: additive combinatorics, arithmetic progressions)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A005487 and check the conjectured growth against them.,https://www.erdosproblems.com/271,
erdos-272,open,needs-theory,mathematics / additive combinatorics,"Let $N\geq 1$. What is the largest $t$ such that there are $A_1,\ldots,A_t\subseteq \{1,\ldots,N\}$ with $A_i\cap A_j$ a non-empty arithmetic progression for all $i\neq j$?","Erdős problems database (status: open; prize: no; tags: additive combinatorics, arithmetic progressions)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/272,
erdos-273,open,needs-theory,mathematics / number theory,Is there a covering system all of whose moduli are of the form $p-1$ for some primes $p\geq 5$?,"Erdős problems database (status: open; prize: no; tags: number theory, covering systems)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/273,
erdos-274,open,needs-theory,mathematics / group theory,If $G$ is a group then can there exist an exact covering of $G$ by more than one cosets of different sizes? (i.e. each element is contained in exactly one of the cosets),"Erdős problems database (status: open; prize: no; tags: group theory, covering systems)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/274,
erdos-276,open,needs-theory,mathematics / number theory,"Is there an infinite Lucas sequence $a_0,a_1,\ldots$ where $a_{n+2}=a_{n+1}+a_n$ for $n\geq 0$ such that all $a_k$ are composite, and yet no integer has a common factor with every term of the sequence?","Erdős problems database (status: open; prize: no; tags: number theory, covering systems)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/276,
erdos-278,open,needs-theory,mathematics / number theory,Let $A=\{n_1<\cdots<n_r\}$ be a finite set of positive integers. What is the maximum density of integers covered by a suitable choice of congruences $a_i\pmod{n_i}$? Is the minimum density achieved when all the $a_i$ are equal?,"Erdős problems database (status: open; prize: no; tags: number theory, covering systems)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/278,
erdos-279,open,needs-theory,mathematics / number theory,Let $k\geq 3$. Is there a choice of congruence classes $a_p\pmod{p}$ for every prime $p$ such that all sufficiently large integers can be written as $a_p+tp$ for some prime $p$ and integer $t\geq k$?,"Erdős problems database (status: open; prize: no; tags: number theory, covering systems, primes)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/279,
erdos-28,open,needs-theory,mathematics / number theory,If $A\subseteq \mathbb{N}$ is such that $A+A$ contains all but finitely many integers then $\limsup 1_A\ast 1_A(n)=\infty$.,"Erdős problems database (status: open; prize: $500; tags: number theory, additive basis)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/28,
erdos-282,open,needs-theory,mathematics / number theory,"Let $A\subseteq \mathbb{N}$ be an infinite set and consider the following greedy algorithm for a rational $x\in (0,1)$: choose the minimal $n\in A$ such that $n\geq 1/x$ and repeat with $x$ replaced by $x-\frac{1}{n}$. If this terminates after finitely many steps then this produces a representation of $x$ as the sum of distinct unit fractions with denominators from $A$. Does this process always terminate if $x$ has odd denominator and $A$ is the set of odd numbers? More generally, for which pairs $x$ and $A$ does this process terminate?","Erdős problems database (status: open; prize: no; tags: number theory, unit fractions)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/282,
erdos-287,open,compute-checkable-small-cases,mathematics / number theory,"Open, but could be disproved with a finite counterexample. Let $k\geq 2$. Is it true that, for any distinct integers $1<n_1<\cdots <n_k$ such that\[1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}\]we must have $\max(n_{i+1}-n_i)\geq 3$?","Erdős problems database (status: falsifiable; prize: no; tags: number theory, unit fractions)",The database marks this as refutable by a finite computation if false: search for a counterexample. Extend OEIS N/A first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/287,
erdos-288,open,needs-theory,mathematics / number theory,"Is it true that there are only finitely many pairs of intervals $I_1,I_2$ such that\[\sum_{n_1\in I_1}\frac{1}{n_1}+\sum_{n_2\in I_2}\frac{1}{n_2}\in \mathbb{N}?\]","Erdős problems database (status: open; prize: no; tags: number theory, unit fractions)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/288,
erdos-289,open,needs-theory,mathematics / number theory,"Is it true that, for all sufficiently large $k$, there exist finite intervals $I_1,\ldots,I_k\subset \mathbb{N}$, distinct, not overlapping or adjacent, with $\lvert I_i\rvert \geq 2$ for $1\leq i\leq k$ such that\[1=\sum_{i=1}^k \sum_{n\in I_i}\frac{1}{n}?\]","Erdős problems database (status: open; prize: no; tags: number theory, unit fractions)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/289,
erdos-291,open,needs-theory,mathematics / number theory,"Let $n\geq 1$ and define $L_n$ to be the least common multiple of $\{1,\ldots,n\}$ and $a_n$ by\[\sum_{1\leq k\leq n}\frac{1}{k}=\frac{a_n}{L_n}.\]Is it true that $(a_n,L_n)=1$ and $(a_n,L_n)>1$ both occur for infinitely many $n$?","Erdős problems database (status: open; prize: no; tags: number theory, unit fractions)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A110566 and check the conjectured growth against them.,https://www.erdosproblems.com/291,
erdos-293,open,needs-theory,mathematics / number theory,Let $k\geq 1$ and let $v(k)$ be the minimal integer which does not appear as some $n_i$ in a solution to\[1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}\]with $1\leq n_1<\cdots <n_k$. Estimate the growth of $v(k)$.,"Erdős problems database (status: open; prize: no; tags: number theory, unit fractions)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/293,
erdos-295,open,needs-theory,mathematics / number theory,Let $N\geq 1$ and let $k(N)$ denote the smallest $k$ such that there exist $N\leq n_1<\cdots <n_k$ with\[1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.\]Is it true that\[\lim_{N\to \infty} k(N)-(e-1)N=\infty?\],"Erdős problems database (status: open; prize: no; tags: number theory, unit fractions)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A192881 and check the conjectured growth against them.,https://www.erdosproblems.com/295,
erdos-3,open,needs-theory,mathematics / number theory,If $A\subseteq \mathbb{N}$ has $\sum_{n\in A}\frac{1}{n}=\infty$ then must $A$ contain arbitrarily long arithmetic progressions?,"Erdős problems database (status: open; prize: $5000; tags: number theory, additive combinatorics, arithmetic progressions)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A003002, A003003, A003004, A003005 and check the conjectured growth against them.",https://www.erdosproblems.com/3,
erdos-30,open,needs-theory,mathematics / number theory,"Let $h(N)$ be the maximum size of a Sidon set in $\{1,\ldots,N\}$. Is it true that, for every $\epsilon>0$,\[h(N) = N^{1/2}+O_\epsilon(N^\epsilon)?\]","Erdős problems database (status: open; prize: $1000; tags: number theory, sidon sets, additive combinatorics)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A143824, A227590, A003022 and check the conjectured growth against them.",https://www.erdosproblems.com/30,
erdos-301,open,needs-theory,mathematics / number theory,"Let $f(N)$ be the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there are no solutions to\[\frac{1}{a}= \frac{1}{b_1}+\cdots+\frac{1}{b_k}\]with distinct $a,b_1,\ldots,b_k\in A$? Estimate $f(N)$. In particular, is it true that $f(N)=(\tfrac{1}{2}+o(1))N$?","Erdős problems database (status: open; prize: no; tags: number theory, unit fractions)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A390394 and check the conjectured growth against them.,https://www.erdosproblems.com/301,
erdos-302,open,needs-theory,mathematics / number theory,"Let $f(N)$ be the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there are no solutions to\[\frac{1}{a}= \frac{1}{b}+\frac{1}{c}\]with distinct $a,b,c\in A$? Estimate $f(N)$. In particular, is $f(N)=(\tfrac{1}{2}+o(1))N$?","Erdős problems database (status: open; prize: no; tags: number theory, unit fractions)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A390395 and check the conjectured growth against them.,https://www.erdosproblems.com/302,
erdos-304,open,needs-theory,mathematics / number theory,"For integers $1\leq a<b$ let $N(a,b)$ denote the minimal $k$ such that there exist integers $1<n_1<\cdots<n_k$ with\[\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.\]Estimate $N(b)=\max_{1\leq a<b}N(a,b)$. Is it true that $N(b) \ll \log\log b$?","Erdős problems database (status: open; prize: no; tags: number theory, unit fractions)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A097847, A097849 and check the conjectured growth against them.",https://www.erdosproblems.com/304,
erdos-306,open,needs-theory,mathematics / number theory,"Let $a/b\in \mathbb{Q}_{>0}$ with $b$ squarefree. Are there integers $1<n_1<\cdots<n_k$, each the product of two distinct primes, such that\[\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}?\]","Erdős problems database (status: open; prize: no; tags: number theory, unit fractions)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/306,
erdos-307,open,compute-checkable-small-cases,mathematics / number theory,"Open, but could be proved with a finite example. Are there two finite sets of primes $P,Q$ such that\[1=\left(\sum_{p\in P}\frac{1}{p}\right)\left(\sum_{q\in Q}\frac{1}{q}\right)?\]","Erdős problems database (status: verifiable; prize: no; tags: number theory, unit fractions)",The database marks this as provable by a finite computation if true: run the verification. Extend OEIS N/A first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/307,
erdos-311,open,needs-theory,mathematics / number theory,"Let $\delta(N)$ be the minimal non-zero value of $\lvert 1-\sum_{n\in A}\frac{1}{n}\rvert$ as $A$ ranges over all subsets of $\{1,\ldots,N\}$. Is it true that\[\delta(N)=e^{-(c+o(1))N}\]for some constant $c\in (0,1)$?","Erdős problems database (status: open; prize: no; tags: number theory, unit fractions)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/311,
erdos-312,open,needs-theory,mathematics / number theory,"Does there exist some $c>0$ such that, for any $K>1$, whenever $A$ is a sufficiently large finite multiset of positive integers with $\sum_{n\in A}\frac{1}{n}>K$ there exists some $S\subseteq A$ such that\[1-e^{-cK} < \sum_{n\in S}\frac{1}{n}\leq 1?\]","Erdős problems database (status: open; prize: no; tags: number theory, unit fractions)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/312,
erdos-313,open,needs-theory,mathematics / number theory,"Are there infinitely many solutions to\[\frac{1}{p_1}+\cdots+\frac{1}{p_k}=1-\frac{1}{m},\]where $m\geq 2$ is an integer and $p_1<\cdots<p_k$ are distinct primes?","Erdős problems database (status: open; prize: no; tags: number theory, unit fractions)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A054377 and check the conjectured growth against them.,https://www.erdosproblems.com/313,
erdos-317,open,needs-theory,mathematics / number theory,"Is there some constant $c>0$ such that for every $n\geq 1$ there exists some $\delta_k\in \{-1,0,1\}$ for $1\leq k\leq n$ with\[0< \left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert < \frac{c}{2^n}?\]Is it true that for sufficiently large $n$, for any $\delta_k\in \{-1,0,1\}$,\[\left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert > \frac{1}{[1,\ldots,n]}\]whenever the left-hand side is not zero?","Erdős problems database (status: open; prize: no; tags: number theory, unit fractions)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/317,
erdos-319,open,needs-theory,mathematics / number theory,"What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there is a function $\delta:A\to \{-1,1\}$ such that\[\sum_{n\in A}\frac{\delta_n}{n}=0\]and\[\sum_{n\in A'}\frac{\delta_n}{n}\neq 0\]for all non-empty $A'\subsetneq A$?","Erdős problems database (status: open; prize: no; tags: number theory, unit fractions)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/319,
erdos-32,open,needs-theory,mathematics / number theory,"Is there a set $A\subset\mathbb{N}$ such that\[\lvert A\cap\{1,\ldots,N\}\rvert = o((\log N)^2)\]and such that every large integer can be written as $p+a$ for some prime $p$ and $a\in A$? Can the bound $O(\log N)$ be achieved? Must such an $A$ satisfy\[\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{\log N}> 1?\]","Erdős problems database (status: open; prize: no; tags: number theory, additive basis)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/32,
erdos-322,open,needs-theory,mathematics / number theory,"Let $k\geq 3$ and $A\subset \mathbb{N}$ be the set of $k$th powers. What is the order of growth of $1_A^{(k)}(n)$, i.e. the number of representations of $n$ as the sum of $k$ many $k$th powers? Does there exist some $c>0$ and infinitely many $n$ such that\[1_A^{(k)}(n) >n^c?\]","Erdős problems database (status: open; prize: no; tags: number theory, powers)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A025456, A025418 and check the conjectured growth against them.",https://www.erdosproblems.com/322,
erdos-323,open,needs-theory,mathematics / number theory,"Let $1\leq m\leq k$ and $f_{k,m}(x)$ denote the number of integers $\leq x$ which are the sum of $m$ many nonnegative $k$th powers. Is it true that\[f_{k,k}(x) \gg_\epsilon x^{1-\epsilon}\]for all $\epsilon>0$? Is it true that if $m<k$ then\[f_{k,m}(x) \gg x^{m/k}\]for sufficiently large $x$?","Erdős problems database (status: open; prize: no; tags: number theory, powers)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A004825, A004831, A004832, A004833, A004842, A004843, A004844, A004845, A004857, A004869, possible and check the conjectured growth against them.",https://www.erdosproblems.com/323,
erdos-324,open,needs-theory,mathematics / number theory,Does there exist a polynomial $f(x)\in\mathbb{Z}[x]$ such that all the sums $f(a)+f(b)$ with $a<b$ nonnegative integers are distinct?,"Erdős problems database (status: open; prize: no; tags: number theory, powers)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/324,
erdos-325,open,needs-theory,mathematics / number theory,"Let $k\geq 3$ and $f_{k,3}(x)$ denote the number of integers $\leq x$ which are the sum of three nonnegative $k$th powers. Is it true that\[f_{k,3}(x) \gg x^{3/k}\]or even $\gg_\epsilon x^{3/k-\epsilon}$?","Erdős problems database (status: open; prize: no; tags: number theory, powers)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A004825, A004832, A004843, A004854, A004865, possible and check the conjectured growth against them.",https://www.erdosproblems.com/325,
erdos-326,open,needs-theory,mathematics / number theory,"Does there exist $A=\{a_1<a_2<\cdots\}\subset \mathbb{N}$ which is a minimal basis of order $2$ (i.e. every large integer is the sum of $2$ elements from $A$, and no proper subset of $A$ has this property), such that\[\lim_{k\to \infty}\frac{a_k}{k^2}=c\]for some $c\neq 0$?","Erdős problems database (status: open; prize: no; tags: number theory, additive basis)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/326,
erdos-327,open,needs-theory,mathematics / number theory,"Suppose $A\subseteq \{1,\ldots,N\}$ is such that if $a,b\in A$ and $a\neq b$ then $a+b\nmid ab$. Can $A$ be 'substantially more' than the odd numbers? What if $a,b\in A$ with $a\neq b$ implies $a+b\nmid 2ab$? Must $\lvert A\rvert=o(N)$?","Erdős problems database (status: open; prize: no; tags: number theory, unit fractions)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A384927 and check the conjectured growth against them.,https://www.erdosproblems.com/327,
erdos-329,open,needs-theory,mathematics / number theory,"Suppose $A\subseteq \mathbb{N}$ is a Sidon set. How large can\[\limsup_{N\to \infty}\frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}}\]be?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/329,
erdos-33,open,needs-theory,mathematics / number theory,"Let $A\subset\mathbb{N}$ be such that every large integer can be written as $n^2+a$ for some $a\in A$ and $n\geq 0$. What is the smallest possible value of\[\limsup \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}?\]Is\[\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}>1?\]","Erdős problems database (status: open; prize: no; tags: number theory, additive basis)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/33,
erdos-332,open,needs-theory,mathematics / number theory,"Let $A\subseteq \mathbb{N}$ and $D(A)$ be the set of those numbers which occur infinitely often as $a_1-a_2$ with $a_1,a_2\in A$. What conditions on $A$ are sufficient to ensure $D(A)$ has bounded gaps?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/332,
erdos-334,open,needs-theory,mathematics / number theory,"Find the best function $f(n)$ such that every $n$ can be written as $n=a+b$ where both $a,b$ are $f(n)$-smooth (that is, are not divisible by any prime $p>f(n)$.)",Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A062241, A045535 and check the conjectured growth against them.",https://www.erdosproblems.com/334,
erdos-335,open,needs-theory,mathematics / number theory,"Let $d(A)$ denote the density of $A\subseteq \mathbb{N}$. Characterise those $A,B\subseteq \mathbb{N}$ with positive density such that\[d(A+B)=d(A)+d(B).\]","Erdős problems database (status: open; prize: no; tags: number theory, additive combinatorics)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/335,
erdos-336,open,needs-theory,mathematics / number theory,For $r\geq 2$ let $h(r)$ be the maximal finite $k$ such that there exists a basis $A\subseteq \mathbb{N}$ of order $r$ (so every large integer is the sum of at most $r$ integers from $A$) and exact order $k$ (so every large integer is the sum of exactly $k$ integers from $A$). Find the value of\[\lim_r \frac{h(r)}{r^2}.\],"Erdős problems database (status: open; prize: no; tags: number theory, additive basis)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/336,
erdos-338,open,needs-theory,mathematics / number theory,The restricted order of a basis is the least integer $t$ (if it exists) such that every large integer is the sum of at most $t$ distinct summands from $A$. What are necessary and sufficient conditions that this exists? Can it be bounded (when it exists) in terms of the order of the basis? What are necessary and sufficient conditions that this is equal to the order of the basis?,"Erdős problems database (status: open; prize: no; tags: number theory, additive basis)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/338,
erdos-340,open,needs-theory,mathematics / number theory,"Let $A=\{1,2,4,8,13,21,31,45,66,81,97,\ldots\}$ be the greedy Sidon sequence: we begin with $1$ and iteratively include the next smallest integer that preserves the Sidon property (i.e. there are no non-trivial solutions to $a+b=c+d$). What is the order of growth of $A$? Is it true that\[\lvert A\cap \{1,\ldots,N\}\rvert \gg N^{1/2-\epsilon}\]for all $\epsilon>0$ and large $N$?","Erdős problems database (status: open; prize: no; tags: number theory, additive combinatorics, sidon sets)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A080200, A005282 and check the conjectured growth against them.",https://www.erdosproblems.com/340,
erdos-341,open,needs-theory,mathematics / number theory,"Let $A=\{a_1<\cdots<a_k\}$ be a finite set of positive integers and extend it to an infinite sequence $\overline{A}=\{a_1<a_2<\cdots \}$ by defining $a_{n+1}$ for $n\geq k$ to be the least integer exceeding $a_n$ which is not of the form $a_i+a_j$ with $i,j\leq n$. Is it true that the sequence of differences $a_{m+1}-a_m$ is eventually periodic?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/341,
erdos-342,open,needs-theory,mathematics / number theory,"With $a_1=1$ and $a_2=2$ let $a_{n+1}$ for $n\geq 2$ be the least integer $>a_n$ which can be expressed uniquely as $a_i+a_j$ for $i<j\leq n$. What can be said about this sequence? Do infinitely many pairs $a,a+2$ occur? Does this sequence eventually have periodic differences? Is the density $0$?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A002858 and check the conjectured growth against them.,https://www.erdosproblems.com/342,
erdos-345,open,needs-theory,mathematics / number theory,"Let $A\subseteq \mathbb{N}$ be a complete sequence, and define the threshold of completeness $T(A)$ to be the least integer $m$ such that all $n\geq m$ are in\[P(A) = \left\{\sum_{n\in B}n : B\subseteq A\textrm{ finite }\right\}\](the existence of $T(A)$ is guaranteed by completeness). Is it true that there are infinitely many $k$ such that $T(n^k)>T(n^{k+1})$?","Erdős problems database (status: open; prize: no; tags: number theory, complete sequences)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A001661 and check the conjectured growth against them.,https://www.erdosproblems.com/345,
erdos-348,open,needs-theory,mathematics / number theory,"For what values of $0\leq m<n$ is there a complete sequence $A=\{a_1\leq a_2\leq \cdots\}$ of integers such that $A$ remains complete after removing any $m$ elements, but $A$ is not complete after removing any $n$ elements?","Erdős problems database (status: open; prize: no; tags: number theory, complete sequences)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/348,
erdos-349,open,needs-theory,mathematics / number theory,"For what values of $t,\alpha \in (0,\infty)$ is the sequence $\lfloor t\alpha^n\rfloor$ complete (that is, all sufficiently large integers are the sum of distinct integers of the form $\lfloor t\alpha^n\rfloor$)?","Erdős problems database (status: open; prize: no; tags: number theory, complete sequences)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/349,
erdos-352,open,needs-theory,mathematics / geometry,Is there some $c>0$ such that every measurable $A\subseteq \mathbb{R}^2$ of measure $\geq c$ contains the vertices of a triangle of area 1?,Erdős problems database (status: open; prize: no; tags: geometry),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/352,
erdos-354,open,needs-theory,mathematics / number theory,"Let $\alpha,\beta\in \mathbb{R}_{>0}$ such that $\alpha/\beta$ is irrational. Is the multiset\[\{ \lfloor \alpha\rfloor,\lfloor 2\alpha\rfloor,\lfloor 4\alpha\rfloor,\ldots\}\cup \{ \lfloor \beta\rfloor,\lfloor 2\beta\rfloor,\lfloor 4\beta\rfloor,\ldots\}\]complete? That is, can all sufficiently large natural numbers $n$ be written as\[n=\sum_{s\in S}\lfloor 2^s\alpha\rfloor+\sum_{t\in T}\lfloor 2^t\beta\rfloor\]for some finite $S,T\subset \mathbb{N}$? What if $2$ is replaced by some $\gamma\in(1,2)$?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/354,
erdos-357,open,needs-theory,mathematics / number theory,Let $1\leq a_1<\cdots <a_k\leq n$ be integers such that all sums of the shape $\sum_{u\leq i\leq v}a_i$ are distinct. Let $f(n)$ be the maximal such $k$. How does $f(n)$ grow? Is $f(n)=o(n)$?,Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A364132, A364153, possible and check the conjectured growth against them.",https://www.erdosproblems.com/357,
erdos-359,open,needs-theory,mathematics / number theory,"Let $a_1<a_2<\cdots$ be an infinite sequence of integers such that $a_1=n$ and $a_{i+1}$ is the least integer which is not a sum of consecutive earlier $a_j$s. What can be said about the density of this sequence? In particular, in the case $n=1$, can one prove that $a_k/k\to \infty$ and $a_k/k^{1+c}\to 0$ for any $c>0$?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A002048 and check the conjectured growth against them.,https://www.erdosproblems.com/359,
erdos-36,open,needs-theory,mathematics / number theory,"Find the optimal constant $c>0$ such that the following holds. For all sufficiently large $N$, if $A\sqcup B=\{1,\ldots,2N\}$ is a partition into two equal parts, so that $\lvert A\rvert=\lvert B\rvert=N$, then there is some $x$ such that the number of solutions to $a-b=x$ with $a\in A$ and $b\in B$ is at least $cN$.","Erdős problems database (status: open; prize: no; tags: number theory, additive combinatorics)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A393584, possible and check the conjectured growth against them.",https://www.erdosproblems.com/36,
erdos-361,open,needs-theory,mathematics / number theory,"Let $c>0$ and $n$ be some large integer. What is the size of the largest $A\subseteq \{1,\ldots,\lfloor cn\rfloor\}$ such that $n$ is not a sum of a subset of $A$? Does this depend on $n$ in an irregular way?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/361,
erdos-364,open,compute-checkable-small-cases,mathematics / number theory,"Open, but could be proved with a finite example. Are there any triples of consecutive positive integers all of which are powerful (i.e. if $p\mid n$ then $p^2\mid n$)?",Erdős problems database (status: verifiable; prize: no; tags: number theory),"The database marks this as provable by a finite computation if true: run the verification. Extend OEIS A060355, A076445 first; a new term is a checkable contribution even without a resolution.",https://www.erdosproblems.com/364,
erdos-365,open,needs-theory,mathematics / number theory,"Do all pairs of consecutive powerful numbers $n$ and $n+1$ come from solutions to Pell equations ? In other words, must either $n$ or $n+1$ be a square? Is the number of such $n\leq x$ bounded by $(\log x)^{O(1)}$?",Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A060355, A060859, A175155 and check the conjectured growth against them.",https://www.erdosproblems.com/365,
erdos-366,open,compute-checkable-small-cases,mathematics / number theory,"Open, but could be proved with a finite example. Are there any $2$-full $n$ such that $n+1$ is $3$-full? That is, if $p\mid n$ then $p^2\mid n$ and if $p\mid n+1$ then $p^3\mid n+1$.",Erdős problems database (status: verifiable; prize: no; tags: number theory),The database marks this as provable by a finite computation if true: run the verification. Extend OEIS A060355 first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/366,
erdos-367,open,needs-theory,mathematics / number theory,"Let $B_2(n)$ be the $2$-full part of $n$ (that is, $B_2(n)=n/n'$ where $n'$ is the product of all primes that divide $n$ exactly once). Is it true that, for every fixed $k\geq 1$,\[\prod_{n\leq m<n+k}B_2(m) \ll n^{2+o(1)}?\]Or perhaps even $\ll_k n^2$?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A057521 and check the conjectured growth against them.,https://www.erdosproblems.com/367,
erdos-368,open,needs-theory,mathematics / number theory,How large is the largest prime factor of $n(n+1)$?,Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A074399 and check the conjectured growth against them.,https://www.erdosproblems.com/368,
erdos-371,open,needs-theory,mathematics / number theory,Let $P(n)$ denote the largest prime factor of $n$. Show that the set of $n$ with $P(n)<P(n+1)$ has density $1/2$.,Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A070089 and check the conjectured growth against them.,https://www.erdosproblems.com/371,
erdos-373,open,needs-theory,mathematics / number theory,"Show that the equation\[n! = a_1!a_2!\cdots a_k!,\]with $n-1>a_1\geq a_2\geq \cdots \geq a_k\geq 2$, has only finitely many solutions.","Erdős problems database (status: open; prize: no; tags: number theory, factorials)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A003135 and check the conjectured growth against them.,https://www.erdosproblems.com/373,
erdos-374,open,needs-theory,mathematics / number theory,"For any $m\in \mathbb{N}$, let $F(m)$ be the minimal $k\geq 2$ (if it exists) such that there are $a_1<\cdots <a_k=m$ with $a_1!\cdots a_k!$ a square. Let $D_k=\{ m : F(m)=k\}$. What is the order of growth of $\lvert D_k\cap\{1,\ldots,n\}\rvert$ for $3\leq k\leq 6$? For example, is it true that $\lvert D_6\cap \{1,\ldots,n\}\rvert \gg n$?",Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A388851, A387184, A389117, A389148 and check the conjectured growth against them.",https://www.erdosproblems.com/374,
erdos-375,open,compute-checkable-small-cases,mathematics / number theory,"Open, but could be disproved with a finite counterexample. Is it true that for any $n,k\geq 1$, if $n+1,\ldots,n+k$ are all composite then there are distinct primes $p_1,\ldots,p_k$ such that $p_i\mid n+i$ for $1\leq i\leq k$?",Erdős problems database (status: falsifiable; prize: no; tags: number theory),The database marks this as refutable by a finite computation if false: search for a counterexample. Extend OEIS N/A first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/375,
erdos-376,open,needs-theory,mathematics / number theory,Are there infinitely many $n$ such that $\binom{2n}{n}$ is coprime to $105$?,"Erdős problems database (status: open; prize: no; tags: number theory, binomial coefficients, base representations)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A030979 and check the conjectured growth against them.,https://www.erdosproblems.com/376,
erdos-377,open,needs-theory,mathematics / number theory,Is there some absolute constant $C>0$ such that\[\sum_{p\leq n}1_{p\nmid \binom{2n}{n}}\frac{1}{p}\leq C\]for all $n$ (where the summation is restricted to primes $p\leq n$)?,"Erdős problems database (status: open; prize: no; tags: number theory, binomial coefficients)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/377,
erdos-382,open,needs-theory,mathematics / number theory,Let $u\leq v$ be such that the largest prime dividing $\prod_{u\leq m\leq v}m$ appears with exponent at least $2$. Is it true that $v-u=v^{o(1)}$? Can $v-u$ be arbitrarily large?,Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A388850 and check the conjectured growth against them.,https://www.erdosproblems.com/382,
erdos-383,open,needs-theory,mathematics / number theory,Is it true that for every $k$ there are infinitely many primes $p$ such that the largest prime divisor of\[\prod_{0\leq i\leq k}(p^2+i)\]is $p$?,Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/383,
erdos-385,open,needs-theory,mathematics / number theory,"Let\[F(n) = \max_{\substack{m<n\\ m\textrm{ composite}}} m+p(m),\]where $p(m)$ is the least prime divisor of $m$. Is it true that $F(n)>n$ for all sufficiently large $n$? Does $F(n)-n\to \infty$ as $n\to\infty$?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A322292 and check the conjectured growth against them.,https://www.erdosproblems.com/385,
erdos-386,open,needs-theory,mathematics / number theory,Let $2\leq k\leq n-2$. Can $\binom{n}{k}$ be the product of consecutive primes infinitely often? For example\[\binom{21}{2}=2\cdot 3\cdot 5\cdot 7.\],"Erdős problems database (status: open; prize: no; tags: number theory, binomial coefficients)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A280992 and check the conjectured growth against them.,https://www.erdosproblems.com/386,
erdos-388,open,needs-theory,mathematics / number theory,"Can one classify all solutions of\[\prod_{1\leq i\leq k_1}(m_1+i)=\prod_{1\leq j\leq k_2}(m_2+j)\]where $k_1,k_2>3$ and $m_1+k_1\leq m_2$? Are there only finitely many solutions?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/388,
erdos-389,open,needs-theory,mathematics / number theory,Is it true that for every $n\geq 1$ there is a $k$ such that\[n(n+1)\cdots(n+k-1)\mid (n+k)\cdots (n+2k-1)?\],Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A375071 and check the conjectured growth against them.,https://www.erdosproblems.com/389,
erdos-39,open,needs-theory,mathematics / number theory,"Is there an infinite Sidon set $A\subset \mathbb{N}$ such that\[\lvert A\cap \{1\ldots,N\}\rvert \gg_\epsilon N^{1/2-\epsilon}\]for all $\epsilon>0$?","Erdős problems database (status: open; prize: $500; tags: number theory, sidon sets, additive combinatorics)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/39,
erdos-390,open,needs-theory,mathematics / number theory,(LEAN) No explanation available Let $f(n)$ be the minimal $m$ such that\[n! = a_1\cdots a_k\]with $n< a_1<\cdots <a_k=m$. Is there (and what is it) a constant $c$ such that\[f(n)-2n \sim c\frac{n}{\log n}?\],"Erdős problems database (status: open (lean); prize: no; tags: number theory, factorials)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A193429 and check the conjectured growth against them.,https://www.erdosproblems.com/390,
erdos-393,open,needs-theory,mathematics / number theory,Let $f(n)$ denote the minimal $m\geq 1$ such that\[n! = a_1\cdots a_t\]with $a_1<\cdots <a_t=a_1+m$. What is the behaviour of $f(n)$?,"Erdős problems database (status: open; prize: no; tags: number theory, factorials)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A388302 and check the conjectured growth against them.,https://www.erdosproblems.com/393,
erdos-394,open,needs-theory,mathematics / number theory,"Let $t_k(n)$ denote the least $m$ such that\[n\mid m(m+1)(m+2)\cdots (m+k-1).\]Is it true that\[\sum_{n\leq x}t_2(n)\ll \frac{x^2}{(\log x)^c}\]for some $c>0$? Is it true that, for $k\geq 2$,\[\sum_{n\leq x}t_{k+1}(n) =o\left(\sum_{n\leq x}t_k(n)\right)?\]",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A344005 and check the conjectured growth against them.,https://www.erdosproblems.com/394,
erdos-396,open,needs-theory,mathematics / number theory,Is it true that for every $k$ there exists $n$ such that\[\prod_{0\leq i\leq k}(n-i) \mid \binom{2n}{n}?\],"Erdős problems database (status: open; prize: no; tags: number theory, binomial coefficients)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A375077 and check the conjectured growth against them.,https://www.erdosproblems.com/396,
erdos-398,open,compute-checkable-small-cases,mathematics / number theory,"Open, but could be disproved with a finite counterexample. Are the only solutions to\[n!=x^2-1\]when $n=4,5,7$?","Erdős problems database (status: falsifiable; prize: no; tags: number theory, factorials)","The database marks this as refutable by a finite computation if false: search for a counterexample. Extend OEIS A146968, A141399 first; a new term is a checkable contribution even without a resolution.",https://www.erdosproblems.com/398,
erdos-40,open,needs-theory,mathematics / number theory,"For what functions $g(N)\to \infty$ is it true that\[\lvert A\cap \{1,\ldots,N\}\rvert \gg \frac{N^{1/2}}{g(N)}\]implies $\limsup 1_A\ast 1_A(n)=\infty$?","Erdős problems database (status: open; prize: $500; tags: number theory, additive basis)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/40,
erdos-400,open,needs-theory,mathematics / number theory,"For any $k\geq 2$ let $g_k(n)$ denote the maximum value of\[(a_1+\cdots+a_k)-n\]where $a_1,\ldots,a_k$ are integers such that $a_1!\cdots a_k! \mid n!$. Can one show that\[\sum_{n\leq x}g_k(n) \sim c_k x\log x\]for some constant $c_k$? Is it true that there is a constant $c_k$ such that for almost all $n<x$ we have\[g_k(n)=c_k\log x+o(\log x)?\]","Erdős problems database (status: open; prize: no; tags: number theory, factorials)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/400,
erdos-404,open,needs-theory,mathematics / number theory,"For which integers $a\geq 1$ and primes $p$ is there a finite upper bound on those $k$ such that there are $a=a_1<\cdots<a_n$ with\[p^k \mid (a_1!+\cdots+a_n!)?\]If $f(a,p)$ is the greatest such $k$, how does this function behave? Is there a prime $p$ and an infinite sequence $a_1<a_2<\cdots$ such that if $p^{m_k}$ is the highest power of $p$ dividing $\sum_{i\leq k}a_i!$ then $m_k\to \infty$?","Erdős problems database (status: open; prize: no; tags: number theory, factorials)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/404,
erdos-406,open,needs-theory,mathematics / number theory,Is it true that there are only finitely many powers of $2$ which have only the digits $0$ and $1$ when written in base $3$?,"Erdős problems database (status: open; prize: no; tags: number theory, base representations)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/406,
erdos-408,open,needs-theory,mathematics / number theory,"Let $\phi(n)$ be the Euler totient function and $\phi_k(n)$ be the iterated $\phi$ function, so that $\phi_1(n)=\phi(n)$ and $\phi_k(n)=\phi(\phi_{k-1}(n))$. Let\[f(n) = \min \{ k : \phi_k(n)=1\}.\]Does $f(n)/\log n$ have a distribution function? Is $f(n)/\log n$ almost always constant? What can be said about the largest prime factor of $\phi_k(n)$ when, say, $k=\log\log n$?","Erdős problems database (status: open; prize: no; tags: number theory, iterated functions)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A049108 and check the conjectured growth against them.,https://www.erdosproblems.com/408,
erdos-409,open,needs-theory,mathematics / number theory,How many iterations of $n\mapsto \phi(n)+1$ are needed before a prime is reached? Can infinitely many $n$ reach the same prime? What is the density of $n$ which reach any fixed prime?,"Erdős problems database (status: open; prize: no; tags: number theory, iterated functions)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A039651, A229487 and check the conjectured growth against them.",https://www.erdosproblems.com/409,
erdos-41,open,needs-theory,mathematics / number theory,"Let $A\subset\mathbb{N}$ be an infinite set such that the triple sums $a+b+c$ are all distinct for $a,b,c\in A$ (aside from the trivial coincidences). Is it true that\[\liminf \frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/3}}=0?\]","Erdős problems database (status: open; prize: $500; tags: number theory, sidon sets, additive combinatorics)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/41,
erdos-410,open,needs-theory,mathematics / number theory,"Let $\sigma_1(n)=\sigma(n)$, the sum of divisors function, and $\sigma_k(n)=\sigma(\sigma_{k-1}(n))$. Is it true that for all $n\geq 2$\[\lim_{k\to \infty} \sigma_k(n)^{1/k}=\infty?\]","Erdős problems database (status: open; prize: no; tags: number theory, iterated functions)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A007497, possible and check the conjectured growth against them.",https://www.erdosproblems.com/410,
erdos-411,open,needs-theory,mathematics / number theory,Let $g_1=g(n)=n+\phi(n)$ and $g_k(n)=g(g_{k-1}(n))$. For which $n$ and $r$ is it true that $g_{k+r}(n)=2g_k(n)$ for all large $k$?,"Erdős problems database (status: open; prize: no; tags: number theory, iterated functions)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A383044, possible and check the conjectured growth against them.",https://www.erdosproblems.com/411,
erdos-412,open,needs-theory,mathematics / number theory,"Let $\sigma_1(n)=\sigma(n)$, the sum of divisors function, and $\sigma_k(n)=\sigma(\sigma_{k-1}(n))$. Is it true that, for every $m,n\geq 2$, there exist some $i,j$ such that $\sigma_i(m)=\sigma_j(n)$?","Erdős problems database (status: open; prize: no; tags: number theory, iterated functions)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A007497, A051572 and check the conjectured growth against them.",https://www.erdosproblems.com/412,
erdos-413,open,needs-theory,mathematics / number theory,"Let $\omega(n)$ count the number of distinct primes dividing $n$. Are there infinitely many $n$ such that, for all $m<n$, we have $m+\omega(m) \leq n$? Can one show that there exists an $\epsilon>0$ such that there are infinitely many $n$ where $m+\epsilon \omega(m)\leq n$ for all $m<n$?","Erdős problems database (status: open; prize: no; tags: number theory, iterated functions)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A005236 and check the conjectured growth against them.,https://www.erdosproblems.com/413,
erdos-414,open,needs-theory,mathematics / number theory,"Let $h_1(n)=h(n)=n+\tau(n)$ (where $\tau(n)$ counts the number of divisors of $n$) and $h_k(n)=h(h_{k-1}(n))$. Is it true, for any $m,n$, there exist $i$ and $j$ such that $h_i(m)=h_j(n)$?","Erdős problems database (status: open; prize: no; tags: number theory, iterated functions)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A064491, possible and check the conjectured growth against them.",https://www.erdosproblems.com/414,
erdos-415,open,needs-theory,mathematics / number theory,"For any $n$ let $F(n)$ be the largest $k$ such that any of the $k!$ possible ordering patterns appears in some sequence of $\phi(m+1),\ldots,\phi(m+k)$ with $m+k\leq n$. Is it true that\[F(n)=(c+o(1))\log\log\log n\]for some constant $c$? Is the first pattern which fails to appear always\[\phi(m+1)>\phi(m+2)>\cdots >\phi(m+k)?\]Is it true that the 'natural' ordering which mimics what happens to $\phi(1),\ldots,\phi(k)$ is the most likely to appear?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/415,
erdos-416,open,needs-theory,mathematics / number theory,Let $V(x)$ count the number of $n\leq x$ such that $\phi(m)=n$ is solvable. Does $V(2x)/V(x)\to 2$? Is there an asymptotic formula for $V(x)$?,Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A264810 and check the conjectured growth against them.,https://www.erdosproblems.com/416,
erdos-417,open,needs-theory,mathematics / number theory,Let\[V'(x)=\#\{\phi(m) : 1\leq m\leq x\}\]and\[V(x)=\#\{\phi(m) \leq x : 1\leq m\}.\]Does $\lim V(x)/V'(x)$ exist? Is it $>1$?,Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A264810, A061070 and check the conjectured growth against them.",https://www.erdosproblems.com/417,
erdos-420,open,needs-theory,mathematics / number theory,"If $\tau(n)$ counts the number of divisors of $n$ then let\[F(f,n)=\frac{\tau((n+\lfloor f(n)\rfloor)!)}{\tau(n!)}.\]Is it true that\[\lim_{n\to \infty}F((\log n)^C,n)=\infty\]for large $C$? Is it true that $F(\log n,n)$ is everywhere dense in $(1,\infty)$? More generally, if $f(n)\leq \log n$ is a monotonic function such that $f(n)\to \infty$ as $n\to \infty$, then is $F(f,n)$ everywhere dense?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/420,
erdos-422,open,needs-theory,mathematics / number theory,Let $f(1)=f(2)=1$ and for $n>2$\[f(n) = f(n-f(n-1))+f(n-f(n-2)).\]Does $f(n)$ miss infinitely many integers? What is its behaviour?,Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A005185 and check the conjectured growth against them.,https://www.erdosproblems.com/422,
erdos-423,open,needs-theory,mathematics / number theory,Let $a_1=1$ and $a_2=2$ and for $k\geq 3$ choose $a_k$ to be the least integer $>a_{k-1}$ which is the sum of at least two consecutive terms of the sequence. What is the asymptotic behaviour of this sequence?,Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A005243 and check the conjectured growth against them.,https://www.erdosproblems.com/423,
erdos-424,open,needs-theory,mathematics / number theory,"Let $a_1=2$ and $a_2=3$ and continue the sequence by appending to $a_1,\ldots,a_n$ all possible values of $a_ia_j-1$ with $i\neq j$. Is it true that the set of integers which eventually appear has positive density?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A005244 and check the conjectured growth against them.,https://www.erdosproblems.com/424,
erdos-425,open,needs-theory,mathematics / number theory,"Let $F(n)$ be the maximum possible size of a subset $A\subseteq\{1,\ldots,N\}$ such that the products $ab$ are distinct for all $a<b$. Is there a constant $c$ such that\[F(n)=\pi(n)+(c+o(1))n^{3/4}(\log n)^{-3/2}?\]If $A\subseteq \{1,\ldots,n\}$ is such that all products $a_1\cdots a_r$ are distinct for $a_1<\cdots <a_r$ then is it true that\[\lvert A\rvert \leq \pi(n)+O(n^{\frac{r+1}{2r}})?\]","Erdős problems database (status: open; prize: no; tags: number theory, sidon sets)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/425,
erdos-428,open,needs-theory,mathematics / number theory,"Is there a set $A\subseteq \mathbb{N}$ such that, for infinitely many $n$, all of $n-a$ are prime for all $a\in A$ with $0<a<n$ and\[\liminf\frac{\lvert A\cap [1,x]\rvert}{\pi(x)}>0?\]","Erdős problems database (status: open; prize: no; tags: number theory, primes)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/428,
erdos-430,open,needs-theory,mathematics / number theory,"Fix some integer $n$ and define a decreasing sequence in $[1,n)$ by $a_1=n-1$ and, for $k\geq 2$, letting $a_k$ be the greatest integer in $[1,a_{k-1})$ such that all of the prime factors of $a_k$ are $>n-a_k$. Is it true that, for sufficiently large $n$, not all of this sequence can be prime?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/430,
erdos-431,open,needs-theory,mathematics / number theory,Are there two infinite sets $A$ and $B$ such that $A+B$ agrees with the set of prime numbers up to finitely many exceptions?,"Erdős problems database (status: open; prize: no; tags: number theory, primes)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/431,
erdos-432,open,needs-theory,mathematics / number theory,"Let $A,B\subseteq \mathbb{N}$ be two infinite sets. How dense can $A+B$ be if all elements of $A+B$ are pairwise relatively prime?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/432,
erdos-436,open,needs-theory,mathematics / number theory,"If $p$ is a prime and $k,m\geq 2$ then let $r(k,m,p)$ be the minimal $r$ such that $r,r+1,\ldots,r+m-1$ are all $k$th power residues modulo $p$. Let\[\Lambda(k,m)=\limsup_{p\to \infty} r(k,m,p).\]Is it true that $\Lambda(k,2)$ is finite for all $k$? Is $\Lambda(k,3)$ finite for all odd $k$? How large are they?",Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A000445, possible and check the conjectured growth against them.",https://www.erdosproblems.com/436,
erdos-44,open,needs-theory,mathematics / number theory,"Let $N\geq 1$ and $A\subset \{1,\ldots,N\}$ be a Sidon set. Is it true that, for any $\epsilon>0$, there exist $M$ and $B\subset \{N+1,\ldots,M\}$ (which may depend on $N,A,\epsilon$) such that $A\cup B\subset \{1,\ldots,M\}$ is a Sidon set of size at least $(1-\epsilon)M^{1/2}$?","Erdős problems database (status: open; prize: no; tags: number theory, sidon sets, additive combinatorics)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/44,
erdos-445,open,needs-theory,mathematics / number theory,"Is it true that, for any $c>1/2$, if $p$ is a sufficiently large prime then, for any $n\geq 0$, there exist $a,b\in(n,n+p^c)$ such that $ab\equiv 1\pmod{p}$?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/445,
erdos-450,open,needs-theory,mathematics / number theory,"How large must $y=y(\epsilon,n)$ be such that the number of integers in $(x,x+y)$ with a divisor in $(n,2n)$ is at most $\epsilon y$?","Erdős problems database (status: open; prize: no; tags: number theory, divisors)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/450,
erdos-451,open,needs-theory,mathematics / number theory,"Estimate $n_k$, the smallest integer $>2k$ such that $\prod_{1\leq i\leq k}(n_k-i)$ has no prime factor in $(k,2k)$.",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A386620 and check the conjectured growth against them.,https://www.erdosproblems.com/451,
erdos-452,open,needs-theory,mathematics / number theory,"Let $\omega(n)$ count the number of distinct prime factors of $n$. What is the size of the largest interval $I\subseteq [x,2x]$ such that $\omega(n)>\log\log n$ for all $n\in I$?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/452,
erdos-454,open,needs-theory,mathematics / number theory,"Let\[f(n) = \min_{i<n} (p_{n+i}+p_{n-i}),\]where $p_k$ is the $k$th prime. Is it true that\[\limsup_n (f(n)-2p_n)=\infty?\]","Erdős problems database (status: open; prize: no; tags: number theory, primes)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A389676, A389677 and check the conjectured growth against them.",https://www.erdosproblems.com/454,
erdos-455,open,needs-theory,mathematics / number theory,Let $q_1<q_2<\cdots$ be a sequence of primes such that\[q_{n+1}-q_n\geq q_n-q_{n-1}.\]Must\[\lim_n \frac{q_n}{n^2}=\infty?\],Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/455,
erdos-456,open,needs-theory,mathematics / number theory,Let $p_n$ be the smallest prime $\equiv 1\pmod{n}$ and let $m_n$ be the smallest integer such that $n\mid \phi(m_n)$. Is it true that $m_n<p_n$ for almost all $n$? Does $p_n/m_n\to \infty$ for almost all $n$? Are there infinitely many primes $p$ such that $p-1$ is the only $n$ for which $m_n=p$?,Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A034694, A061026, possible and check the conjectured growth against them.",https://www.erdosproblems.com/456,
erdos-458,open,compute-checkable-small-cases,mathematics / number theory,"Open, but could be disproved with a finite counterexample. Let $[1,\ldots,n]$ denote the least common multiple of $\{1,\ldots,n\}$. Is it true that, for all $k\geq 1$,\[[1,\ldots,p_{k+1}-1]< p_k[1,\ldots,p_k]?\]","Erdős problems database (status: falsifiable; prize: no; tags: number theory, primes)",The database marks this as refutable by a finite computation if false: search for a counterexample. Extend OEIS A056604 first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/458,
erdos-460,open,needs-theory,mathematics / number theory,"Let $a_0=0$ and $a_1=1$, and in general define $a_k$ to be the least integer $>a_{k-1}$ for which $(n-a_k,n-a_i)=1$ for all $0\leq i<k$. Does\[\sum_{0<a_i< n}\frac{1}{a_i}\to \infty\]as $n\to \infty$? What about if we restrict the sum to those $i$ such that $n-a_j$ is divisible by some prime $\leq a_j$, or the complement of such $i$?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/460,
erdos-461,open,needs-theory,mathematics / number theory,"Let $s_t(n)$ be the $t$-smooth component of $n$ - that is, the product of all primes $p$ (with multiplicity) dividing $n$ such that $p<t$. Let $f(n,t)$ count the number of distinct possible values for $s_t(m)$ for $m\in [n+1,n+t]$. Is it true that\[f(n,t)\gg t\](uniformly, for all $t$ and $n$)?","Erdős problems database (status: open; prize: no; tags: number theory, primes)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/461,
erdos-462,open,needs-theory,mathematics / number theory,Let $p(n)$ denote the least prime factor of $n$. There is a constant $c>0$ such that\[\sum_{\substack{n<x\\ n\textrm{ not prime}}}\frac{p(n)}{n}\sim c\frac{x^{1/2}}{(\log x)^2}.\]Is it true that there exists a constant $C>0$ such that\[\sum_{x\leq n\leq x+Cx^{1/2}(\log x)^2}\frac{p(n)}{n} \gg 1\]for all large $x$?,"Erdős problems database (status: open; prize: no; tags: number theory, primes)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A032742, possible and check the conjectured growth against them.",https://www.erdosproblems.com/462,
erdos-463,open,needs-theory,mathematics / number theory,"Is there a function $f$ with $f(n)\to \infty$ as $n\to \infty$ such that, for all large $n$, there is a composite number $m$ such that\[n+f(n)<m<n+p(m)?\](Here $p(m)$ is the least prime factor of $m$.)","Erdős problems database (status: open; prize: no; tags: number theory, primes)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/463,
erdos-467,open,needs-theory,mathematics / number theory,"Prove the following for all large $x$: there is a choice of congruence classes $a_p$ for all primes $p\leq x$ and a decomposition $\{p\leq x\}=A\sqcup B$ into two non-empty sets such that, for all $n<x$, there exist some $p\in A$ and $q\in B$ such that $n\equiv a_p\pmod{p}$ and $n\equiv a_q\pmod{q}$.",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/467,
erdos-468,open,needs-theory,mathematics / number theory,"For any $n$ let $D_n$ be the set of sums of the shape $d_1,d_1+d_2,d_1+d_2+d_3,\ldots$ where $1<d_1<d_2<\cdots$ are the divisors of $n$. What is the size of $D_n\backslash \cup_{m<n}D_m$? If $f(N)$ is the minimal $n$ such that $N\in D_n$ then is it true that $f(N)=o(N)$? Perhaps just for almost all $N$?","Erdős problems database (status: open; prize: no; tags: number theory, divisors)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A167485, A387502, A387503 and check the conjectured growth against them.",https://www.erdosproblems.com/468,
erdos-470,open,needs-theory,mathematics / number theory,"Call $n$ weird if $\sigma(n)\geq 2n$ and $n$ is not pseudoperfect, that is, it is not the sum of any set of its divisors. Are there any odd weird numbers? Are there infinitely many primitive weird numbers, i.e. those such that no proper divisor of $n$ is weird?","Erdős problems database (status: open; prize: $10; tags: number theory, divisors)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A006037, A002975 and check the conjectured growth against them.",https://www.erdosproblems.com/470,
erdos-472,open,needs-theory,mathematics / number theory,Given some initial finite sequence of primes $q_1<\cdots<q_m$ extend it so that $q_{n+1}$ is the smallest prime of the form $q_n+q_i-1$ for $n\geq m$. Is there an initial starting sequence so that the resulting sequence is infinite?,Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A389713, possible and check the conjectured growth against them.",https://www.erdosproblems.com/472,
erdos-474,open,needs-theory,mathematics / set theory,"Open in general, but there exist models of set theory where the result is false. - $100 Under what set theoretic assumptions is it true that $\mathbb{R}^2$ can be $3$-coloured such that, for every uncountable $A\subseteq \mathbb{R}^2$, $A^2$ contains a pair of each colour?","Erdős problems database (status: not provable; prize: $100; tags: set theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/474,
erdos-475,open,compute-checkable-small-cases,mathematics / number theory,"Resolved up to a finite check. Let $p$ be a prime. Given any finite set $A\subseteq \mathbb{F}_p\backslash \{0\}$, is there always a rearrangement $A=\{a_1,\ldots,a_t\}$ such that all partial sums $\sum_{1\leq k\leq m}a_{k}$ are distinct, for all $1\leq m\leq t$?","Erdős problems database (status: decidable; prize: no; tags: number theory, additive combinatorics)",The database marks this as reduced to a finite computation: run it. Extend OEIS N/A first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/475,
erdos-477,open,needs-theory,mathematics / number theory,Is there a polynomial $f:\mathbb{Z}\to \mathbb{Z}$ of degree at least $2$ and a set $A\subset \mathbb{Z}$ such that for any $n\in \mathbb{Z}$ there is exactly one $a\in A$ and $b\in \{ f(k) : k\in\mathbb{Z}\}$ such that $n=a+b$?,Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/477,
erdos-478,open,needs-theory,mathematics / number theory,Let $p$ be a prime and\[A_p = \{ k! \pmod{p} : 1\leq k<p\}.\]Is it true that\[\lvert A_p\rvert \sim (1-\tfrac{1}{e})p?\],"Erdős problems database (status: open; prize: no; tags: number theory, factorials)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A210184 and check the conjectured growth against them.,https://www.erdosproblems.com/478,
erdos-479,open,needs-theory,mathematics / number theory,"Is it true that, for all $k\neq 1$, there are infinitely many $n$ such that $2^n\equiv k\pmod{n}$?",Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A036236, A015919, A050259, A015921, A006521, A006517, A015940 and check the conjectured growth against them.",https://www.erdosproblems.com/479,
erdos-483,open,needs-theory,mathematics / number theory,"Let $f(k)$ be the minimal $N$ such that if $\{1,\ldots,N\}$ is $k$-coloured then there is a monochromatic solution to $a+b=c$. Estimate $f(k)$. In particular, is it true that $f(k) < c^k$ for some constant $c>0$?","Erdős problems database (status: open; prize: no; tags: number theory, additive combinatorics, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A030126 and check the conjectured growth against them.,https://www.erdosproblems.com/483,
erdos-486,open,needs-theory,mathematics / number theory,"Let $A\subseteq \mathbb{N}$, and for each $n\in A$ choose some $X_n\subseteq \mathbb{Z}/n\mathbb{Z}$. Let\[B = \{ m\in \mathbb{N} : m\not\in X_n\pmod{n}\textrm{ for all }n\in A\textrm{ with }m>n\}.\]Must $B$ have a logarithmic density, i.e. is it true that\[\lim_{x\to \infty} \frac{1}{\log x}\sum_{\substack{m\in B\\ m<x}}\frac{1}{m}\]exists?","Erdős problems database (status: open; prize: no; tags: number theory, primitive sets)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/486,
erdos-488,open,compute-checkable-small-cases,mathematics / number theory,"Open, but could be disproved with a finite counterexample. Let $A$ be a finite set and\[B=\{ n \geq 1 : a\mid n\textrm{ for some }a\in A\}.\]Is it true that, for every $m>n\geq \max(A)$,\[\frac{\lvert B\cap [1,m]\rvert }{m}< 2\frac{\lvert B\cap [1,n]\rvert}{n}?\]",Erdős problems database (status: falsifiable; prize: no; tags: number theory),The database marks this as refutable by a finite computation if false: search for a counterexample. Extend OEIS N/A first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/488,
erdos-489,open,needs-theory,mathematics / number theory,"Let $A\subseteq \mathbb{N}$ be a set such that $\lvert A\cap [1,x]\rvert=o(x^{1/2})$. Let\[B=\{ n\geq 1 : a\nmid n\textrm{ for all }a\in A\}.\]If $B=\{b_1<b_2<\cdots\}$ then is it true that\[\lim \frac{1}{x}\sum_{b_i<x}(b_{i+1}-b_i)^2\]exists (and is finite)?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/489,
erdos-495,open,needs-theory,mathematics / diophantine approximation,"Let $\alpha,\beta \in \mathbb{R}$. Is it true that\[\liminf_{n\to \infty} n \| n\alpha \| \| n\beta\| =0\]where $\|x\|$ is the distance from $x$ to the nearest integer?","Erdős problems database (status: open; prize: no; tags: diophantine approximation, number theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/495,
erdos-5,open,needs-theory,mathematics / number theory,Let $C\geq 0$. Is there an infinite sequence of $n_i$ such that\[\lim_{i\to \infty}\frac{p_{n_i+1}-p_{n_i}}{\log n_i}=C?\],"Erdős problems database (status: open; prize: no; tags: number theory, primes)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A001223 and check the conjectured growth against them.,https://www.erdosproblems.com/5,
erdos-50,open,needs-theory,mathematics / number theory,"Schoenberg proved that for every $c\in [0,1]$ the density of\[\{ n\in \mathbb{N} : \phi(n)<cn\}\]exists. Let this density be denoted by $f(c)$. Is it true that there are no $x$ such that $f'(x)$ exists and is positive?",Erdős problems database (status: open; prize: $250; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/50,
erdos-500,open,needs-theory,mathematics / graph theory,"What is $\mathrm{ex}_3(n,K_4^3)$? That is, the largest number of $3$-edges which can placed on $n$ vertices so that there exists no $K_4^3$, a set of 4 vertices which is covered by all 4 possible $3$-edges.","Erdős problems database (status: open; prize: $500; tags: graph theory, hypergraphs, turan number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A140462 and check the conjectured growth against them.,https://www.erdosproblems.com/500,
erdos-501,open,needs-theory,mathematics / combinatorics,"For every $x\in\mathbb{R}$ let $A_x\subset \mathbb{R}$ be a bounded set with outer measure $<1$. Must there exist an infinite independent set, that is, some infinite $X\subseteq \mathbb{R}$ such that $x\not\in A_y$ for all $x\neq y\in X$? If the sets $A_x$ are closed and have measure $<1$, then must there exist an independent set of size $3$?","Erdős problems database (status: open; prize: no; tags: combinatorics, set theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/501,
erdos-503,open,needs-theory,mathematics / geometry,"What is the size of the largest $A\subseteq \mathbb{R}^d$ such that every three points from $A$ determine an isosceles triangle? That is, for any three points $x,y,z$ from $A$, at least two of the distances $\lvert x-y\rvert,\lvert y-z\rvert,\lvert x-z\rvert$ are equal.","Erdős problems database (status: open; prize: no; tags: geometry, distances)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A175769 and check the conjectured growth against them.,https://www.erdosproblems.com/503,
erdos-506,open,compute-checkable-small-cases,mathematics / geometry,"Resolved up to a finite check. What is the minimum number of circles determined by any $n$ points in $\mathbb{R}^2$, not all on a circle?",Erdős problems database (status: decidable; prize: no; tags: geometry),The database marks this as reduced to a finite computation: run it. Extend OEIS possible first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/506,
erdos-507,open,needs-theory,mathematics / geometry,Let $\alpha(n)$ be such that every set of $n$ points in the unit disk contains three points which determine a triangle of area at most $\alpha(n)$. Estimate $\alpha(n)$.,Erdős problems database (status: open; prize: no; tags: geometry),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/507,
erdos-508,open,needs-theory,mathematics / geometry,"What is the chromatic number of the plane? That is, what is the smallest number of colours required to colour $\mathbb{R}^2$ such that no two points of the same colour are distance $1$ apart?","Erdős problems database (status: open; prize: no; tags: geometry, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/508,
erdos-509,open,needs-theory,mathematics / analysis,Let $f(z)\in\mathbb{C}[z]$ be a monic non-constant polynomial. Can the set\[\{ z\in \mathbb{C} : \lvert f(z)\rvert \leq 1\}\]be covered by a set of circles the sum of whose radii is $\leq 2$?,Erdős problems database (status: open; prize: no; tags: analysis),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/509,
erdos-51,open,needs-theory,mathematics / number theory,"Is there an infinite set $A\subset \mathbb{N}$ such that for every $a\in A$ there is an integer $n$ such that $\phi(n)=a$, and yet if $n_a$ is the smallest such integer then $n_a/a\to \infty$ as $a\to\infty$?",Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A002202, A014197 and check the conjectured growth against them.",https://www.erdosproblems.com/51,
erdos-510,open,needs-theory,mathematics / analysis,If $A\subset \mathbb{Z}$ is a finite set of size $N$ then is there some absolute constant $c>0$ and $\theta$ such that\[\sum_{n\in A}\cos(n\theta) < -cN^{1/2}?\],Erdős problems database (status: open; prize: no; tags: analysis),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/510,
erdos-513,open,needs-theory,mathematics / analysis,Let $f=\sum_{n=0}^\infty a_nz^n$ be a transcendental entire function. What is the greatest possible value of\[\liminf_{r\to \infty} \frac{\max_n\lvert a_nr^n\rvert}{\max_{\lvert z\rvert=r}\lvert f(z)\rvert}?\],Erdős problems database (status: open; prize: no; tags: analysis),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/513,
erdos-514,open,needs-theory,mathematics / analysis,"Let $f(z)$ be an entire transcendental function. Does there exist a path $L$ so that, for every $n$,\[\lvert f(z)/z^n\rvert \to \infty\]as $z\to \infty$ along $L$? Can the length of this path be estimated in terms of $M(r)=\max_{\lvert z\rvert=r}\lvert f(z)\rvert$? Does there exist a path along which $\lvert f(z)\rvert$ tends to $\infty$ faster than a fixed function of $M(r)$ (such that $M(r)^\epsilon$)?",Erdős problems database (status: open; prize: no; tags: analysis),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/514,
erdos-517,open,needs-theory,mathematics / analysis,Let $f(z)=\sum_{k=1}^\infty a_kz^{n_k}$ be an entire function (with $a_k\neq 0$ for all $k\geq 1$). Is it true that if $n_k/k\to \infty$ then $f(z)$ assumes every value infinitely often?,Erdős problems database (status: open; prize: no; tags: analysis),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/517,
erdos-52,open,needs-theory,mathematics / number theory,"Let $A$ be a finite set of integers. Is it true that for every $\epsilon>0$\[\max( \lvert A+A\rvert,\lvert AA\rvert)\gg_\epsilon \lvert A\rvert^{2-\epsilon}?\]","Erdős problems database (status: open; prize: $250; tags: number theory, additive combinatorics)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A263996 and check the conjectured growth against them.,https://www.erdosproblems.com/52,
erdos-520,open,needs-theory,mathematics / number theory,"Let $f$ be a Rademacher multiplicative function: a random $\{-1,0,1\}$-valued multiplicative function, where for each prime $p$ we independently choose $f(p)\in \{-1,1\}$ uniformly at random, and for square-free integers $n$ we extend $f(p_1\cdots p_r)=f(p_1)\cdots f(p_r)$ (and $f(n)=0$ if $n$ is not squarefree). Does there exist some constant $c>0$ such that, almost surely,\[\limsup_{N\to \infty}\frac{\sum_{m\leq N}f(m)}{\sqrt{N\log\log N}}=c?\]","Erdős problems database (status: open; prize: no; tags: number theory, probability)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/520,
erdos-521,open,needs-theory,mathematics / analysis,"Let $(\epsilon_k)_{k\geq 0}$ be independently uniformly chosen at random from $\{-1,1\}$. If $R_n$ counts the number of real roots of $f_n(z)=\sum_{0\leq k\leq n}\epsilon_k z^k$ then is it true that, almost surely,\[\lim_{n\to \infty}\frac{R_n}{\log n}=\frac{2}{\pi}?\]","Erdős problems database (status: open; prize: no; tags: analysis, polynomials, probability)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/521,
erdos-522,open,needs-theory,mathematics / analysis,"Let $f(z)=\sum_{0\leq k\leq n} \epsilon_k z^k$ be a random polynomial, where $\epsilon_k\in \{-1,1\}$ independently uniformly at random for $0\leq k\leq n$. Is it true that, if $R_n$ is the number of roots of $f(z)$ in $\{ z\in \mathbb{C} : \lvert z\rvert \leq 1\}$, then\[\frac{R_n}{n/2}\to 1\]almost surely?","Erdős problems database (status: open; prize: no; tags: analysis, polynomials, probability)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/522,
erdos-524,open,needs-theory,mathematics / analysis,"For any $t\in (0,1)$ let $t=\sum_{k=1}^\infty \epsilon_k(t)2^{-k}$ (where $\epsilon_k(t)\in \{0,1\}$). What is the correct order of magnitude (for almost all $t\in(0,1)$) for\[M_n(t)=\max_{x\in [-1,1]}\left\lvert \sum_{k\leq n}(-1)^{\epsilon_k(t)}x^k\right\rvert?\]","Erdős problems database (status: open; prize: no; tags: analysis, probability, polynomials)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/524,
erdos-528,open,needs-theory,mathematics / geometry,"Let $f(n,k)$ count the number of self-avoiding walks of $n$ steps (beginning at the origin) in $\mathbb{Z}^k$ (i.e. those walks which do not intersect themselves). Determine\[C_k=\lim_{n\to\infty}f(n,k)^{1/n}.\]",Erdős problems database (status: open; prize: no; tags: geometry),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A387897, A156816 and check the conjectured growth against them.",https://www.erdosproblems.com/528,
erdos-529,open,needs-theory,mathematics / geometry,"Let $d_k(n)$ be the expected distance from the origin after taking $n$ random steps from the origin in $\mathbb{Z}^k$ (conditional on no self intersections) - that is, a self-avoiding walk . Is it true that\[\lim_{n\to \infty}\frac{d_2(n)}{n^{1/2}}= \infty?\]Is it true that\[d_k(n)\ll n^{1/2}\]for $k\geq 3$?","Erdős problems database (status: open; prize: no; tags: geometry, probability)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/529,
erdos-530,open,needs-theory,mathematics / number theory,"Let $\ell(N)$ be maximal such that in any finite set $A\subset \mathbb{R}$ of size $N$ there exists a Sidon subset $S$ of size $\ell(N)$ (i.e. the only solutions to $a+b=c+d$ in $S$ are the trivial ones). Determine the order of $\ell(N)$. In particular, is it true that $\ell(N)\sim N^{1/2}$?","Erdős problems database (status: open; prize: no; tags: number theory, sidon sets)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A143824, possible and check the conjectured growth against them.",https://www.erdosproblems.com/530,
erdos-531,open,needs-theory,mathematics / number theory,"Let $F(k)$ be the minimal $N$ such that if we two-colour $\{1,\ldots,N\}$ there is a set $A$ of size $k$ such that all subset sums $\sum_{a\in S}a$ (for $\emptyset\neq S\subseteq A$) are monochromatic. Estimate $F(k)$.","Erdős problems database (status: open; prize: no; tags: number theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/531,
erdos-535,open,needs-theory,mathematics / number theory,"Let $r\geq 3$, and let $f_r(N)$ denote the size of the largest subset of $\{1,\ldots,N\}$ such that no subset of size $r$ has the same pairwise greatest common divisor between all elements. Estimate $f_r(N)$.",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/535,
erdos-536,open,needs-theory,mathematics / number theory,"Let $f(N)$ be the largest size of $A\subseteq \{1,\ldots,N\}$ with the property that there are no distinct $a,b,c\in A$ such that\[[a,b]=[b,c]=[a,c],\]where $[a,b]$ denotes the least common multiple. Estimate $f(N)$ - in particular, is it true that $f(N)=o(N)$?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/536,
erdos-538,open,needs-theory,mathematics / number theory,"Let $r\geq 2$ and suppose that $A\subseteq\{1,\ldots,N\}$ is such that, for any $m$, there are at most $r$ solutions to $m=pa$ where $p$ is prime and $a\in A$. Give the best possible upper bound for\[\sum_{n\in A}\frac{1}{n}.\]",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/538,
erdos-539,open,needs-theory,mathematics / number theory,"Let $h(n)$ be such that, for any set $A\subseteq \mathbb{N}$ of size $n$, the set\[\left\{ \frac{a}{(a,b)}: a,b\in A\right\}\]has size at least $h(n)$. Estimate $h(n)$.",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/539,
erdos-544,open,needs-theory,mathematics / graph theory,"Show that\[R(3,k+1)-R(3,k)\to\infty\]as $k\to \infty$. Similarly, prove or disprove that\[R(3,k+1)-R(3,k)=o(k).\]","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A000791 and check the conjectured growth against them.,https://www.erdosproblems.com/544,
erdos-545,open,needs-theory,mathematics / graph theory,"Let $G$ be a graph with $m$ edges and no isolated vertices. Is the Ramsey number $R(G)$ maximised when $G$ is 'as complete as possible'? That is, if $m=\binom{n}{2}+t$ edges with $0\leq t<n$ then is\[R(G)\leq R(H),\]where $H$ is the graph formed by connecting a new vertex to $t$ of the vertices of $K_n$?","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A059442, possible and check the conjectured growth against them.",https://www.erdosproblems.com/545,
erdos-547,open,compute-checkable-small-cases,mathematics / graph theory,Resolved up to a finite check. If $T$ is a tree on $n$ vertices then\[R(T) \leq 2n-2.\],"Erdős problems database (status: decidable; prize: no; tags: graph theory, ramsey theory)",The database marks this as reduced to a finite computation: run it. Extend OEIS N/A first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/547,
erdos-548,open,compute-checkable-small-cases,mathematics / graph theory,"Open, but could be disproved with a finite counterexample. - $100 Let $n\geq k+1$. Every graph on $n$ vertices with at least $\frac{k-1}{2}n+1$ edges contains every tree on $k+1$ vertices.",Erdős problems database (status: falsifiable; prize: no; tags: graph theory),The database marks this as refutable by a finite computation if false: search for a counterexample. Extend OEIS N/A first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/548,
erdos-550,open,needs-theory,mathematics / graph theory,"Let $m_1\leq\cdots\leq m_k$ and $n$ be sufficiently large. If $T$ is a tree on $n$ vertices and $G$ is the complete multipartite graph with vertex class sizes $m_1,\ldots,m_k$ then prove that\[R(T,G)\leq (\chi(G)-1)(R(T,K_{m_1,m_2})-1)+m_1.\]","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/550,
erdos-551,open,compute-checkable-small-cases,mathematics / graph theory,"Resolved up to a finite check. Prove that\[R(C_k,K_n)=(k-1)(n-1)+1\]for $k\geq n\geq 3$ (except when $n=k=3$).","Erdős problems database (status: decidable; prize: no; tags: graph theory, ramsey theory)",The database marks this as reduced to a finite computation: run it. Extend OEIS N/A first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/551,
erdos-552,open,needs-theory,mathematics / graph theory,"Determine the Ramsey number\[R(C_4,S_n),\]where $S_n=K_{1,n}$ is the star on $n+1$ vertices. In particular, is it true that, for any $c>0$, there are infinitely many $n$ such that\[R(C_4,S_n)\leq n+\sqrt{n}-c?\]","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A006672 and check the conjectured growth against them.,https://www.erdosproblems.com/552,
erdos-554,open,needs-theory,mathematics / graph theory,Let $R_k(G)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Show that\[\lim_{k\to \infty}\frac{R_k(C_{2n+1})}{R_k(K_3)}=0\]for any $n\geq 2$.,"Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/554,
erdos-555,open,needs-theory,mathematics / graph theory,Let $R_k(G)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Determine the value of\[R_k(C_{2n}).\],"Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A389313, possible and check the conjectured growth against them.",https://www.erdosproblems.com/555,
erdos-556,open,compute-checkable-small-cases,mathematics / graph theory,Resolved up to a finite check. Let $R_3(G)$ denote the minimal $m$ such that if the edges of $K_m$ are $3$-coloured then there must be a monochromatic copy of $G$. Show that\[R_3(C_n) \leq 4n-3.\],"Erdős problems database (status: decidable; prize: no; tags: graph theory, ramsey theory)",The database marks this as reduced to a finite computation: run it. Extend OEIS A389335 first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/556,
erdos-557,open,needs-theory,mathematics / graph theory,Let $R_k(G)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Is it true that\[R_k(T)\leq kn+O(1)\]for any tree $T$ on $n$ vertices?,"Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/557,
erdos-558,open,needs-theory,mathematics / graph theory,"Let $R_k(G)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Determine\[R_k(K_{s,t})\]where $K_{s,t}$ is the complete bipartite graph with $s$ vertices in one component and $t$ in the other.","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/558,
erdos-560,open,needs-theory,mathematics / graph theory,"Let $\hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges such that in any $2$-colouring of the edges of $H$ there is a monochromatic copy of $G$. Determine\[\hat{R}(K_{n,n}),\]where $K_{n,n}$ is the complete bipartite graph with $n$ vertices in each component.","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/560,
erdos-561,open,needs-theory,mathematics / graph theory,"Let $\hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges such that in any $2$-colouring of the edges of $H$ there is a monochromatic copy of $G$. Let $F_1$ and $F_2$ be the union of stars. More precisely, let $F_1=\cup_{i\leq s} K_{1,n_i}$ and $F_2=\cup_{j\leq t} K_{1,m_j}$ with $n_1\geq \cdots \geq n_s\geq 1$ and $m_1\geq \cdots \geq m_t\geq 1$. Prove that\[\hat{R}(F_1,F_2) = \sum_{2\leq k\leq s+t}l_k\]where\[l_k=\max\{n_i+m_j-1 : i+j=k\}.\]","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/561,
erdos-562,open,needs-theory,mathematics / graph theory,"Let $R_r(n)$ denote the $r$-uniform hypergraph Ramsey number: the minimal $m$ such that if we $2$-colour all edges of the complete $r$-uniform hypergraph on $m$ vertices then there must be some monochromatic copy of the complete $r$-uniform hypergraph on $n$ vertices. Prove that, for $r\geq 3$,\[\log_{r-1} R_r(n) \asymp_r n,\]where $\log_{r-1}$ denotes the $(r-1)$-fold iterated logarithm. That is, does $R_r(n)$ grow like\[2^{2^{\cdots n}}\]where the tower of exponentials has height $r-1$?","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory, hypergraphs)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/562,
erdos-563,open,needs-theory,mathematics / graph theory,"Let $F(n,\alpha)$ denote the smallest $m$ such that there exists a $2$-colouring of the edges of $K_n$ so that every $X\subseteq [n]$ with $\lvert X\rvert\geq m$ contains more than $\alpha \binom{\lvert X\rvert}{2}$ many edges of each colour. Prove that, for every $0\leq \alpha< 1/2$,\[F(n,\alpha)\sim c_\alpha\log n\]for some constant $c_\alpha$ depending only on $\alpha$.","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory, hypergraphs)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/563,
erdos-564,open,needs-theory,mathematics / graph theory,Let $R_3(n)$ be the minimal $m$ such that if the edges of the $3$-uniform hypergraph on $m$ vertices are $2$-coloured then there is a monochromatic copy of the complete $3$-uniform hypergraph on $n$ vertices. Is there some constant $c>0$ such that\[R_3(n) \geq 2^{2^{cn}}?\],"Erdős problems database (status: open; prize: $500; tags: graph theory, ramsey theory, hypergraphs)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/564,
erdos-566,open,needs-theory,mathematics / graph theory,"Let $G$ be such that any subgraph on $k$ vertices has at most $2k-3$ edges. Is it true that, if $H$ has $m$ edges and no isolated vertices, then\[R(G,H)\ll m?\]","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/566,
erdos-567,open,needs-theory,mathematics / graph theory,"Let $G$ be either $Q_3$ or $K_{3,3}$ or $H_5$ (the last formed by adding two vertex-disjoint chords to $C_5$). Is it true that, if $H$ has $m$ edges and no isolated vertices, then\[R(G,H)\ll m?\]","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/567,
erdos-568,open,needs-theory,mathematics / graph theory,"Let $G$ be a graph such that $R(G,T_n)\ll n$ for any tree $T_n$ on $n$ vertices and $R(G,K_n)\ll n^2$. Is it true that, for any $H$ with $m$ edges and no isolated vertices,\[R(G,H)\ll m?\]","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/568,
erdos-569,open,needs-theory,mathematics / graph theory,"Let $k\geq 1$. What is the best possible $c_k$ such that\[R(C_{2k+1},H)\leq c_k m\]for any graph $H$ on $m$ edges without isolated vertices?","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/569,
erdos-571,open,needs-theory,mathematics / graph theory,"Show that for any rational $\alpha \in [1,2)$ there exists a bipartite graph $G$ such that\[\mathrm{ex}(n;G)\asymp n^{\alpha}.\]","Erdős problems database (status: open; prize: no; tags: graph theory, turan number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/571,
erdos-572,open,needs-theory,mathematics / graph theory,Show that for $k\geq 3$\[\mathrm{ex}(n;C_{2k})\gg n^{1+\frac{1}{k}}.\],"Erdős problems database (status: open; prize: no; tags: graph theory, turan number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/572,
erdos-573,open,needs-theory,mathematics / graph theory,"Is it true that\[\mathrm{ex}(n;\{C_3,C_4\})\sim (n/2)^{3/2}?\]","Erdős problems database (status: open; prize: no; tags: graph theory, turan number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A006856 and check the conjectured growth against them.,https://www.erdosproblems.com/573,
erdos-576,open,needs-theory,mathematics / graph theory,Let $Q_k$ be the $k$-dimensional hypercube graph (so that $Q_k$ has $2^k$ vertices and $k2^{k-1}$ edges). Determine the behaviour of\[\mathrm{ex}(n;Q_k).\],"Erdős problems database (status: open; prize: no; tags: graph theory, turan number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/576,
erdos-579,open,needs-theory,mathematics / graph theory,"Let $\delta>0$. If $n$ is sufficiently large and $G$ is a graph on $n$ vertices with no $K_{2,2,2}$ and at least $\delta n^2$ edges then $G$ contains an independent set of size $\gg_\delta n$.","Erdős problems database (status: open; prize: no; tags: graph theory, turan number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/579,
erdos-580,open,compute-checkable-small-cases,mathematics / graph theory,Resolved up to a finite check. Let $G$ be a graph on $n$ vertices such that at least $n/2$ vertices have degree at least $n/2$. Must $G$ contain every tree on at most $n/2$ vertices?,Erdős problems database (status: decidable; prize: no; tags: graph theory),The database marks this as reduced to a finite computation: run it. Extend OEIS N/A first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/580,
erdos-583,open,compute-checkable-small-cases,mathematics / graph theory,"Open, but could be disproved with a finite counterexample. Every connected graph on $n$ vertices can be partitioned into at most $\lceil n/2\rceil$ edge-disjoint paths.",Erdős problems database (status: falsifiable; prize: no; tags: graph theory),The database marks this as refutable by a finite computation if false: search for a counterexample. Extend OEIS N/A first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/583,
erdos-584,open,needs-theory,mathematics / graph theory,"Let $G$ be a graph with $n$ vertices and $\delta n^{2}$ edges. Are there subgraphs $H_1,H_2\subseteq G$ such that $H_1$ has $\gg \delta^3n^2$ edges and every two edges in $H_1$ are contained in a cycle of length at most $6$, and furthermore if two edges share a vertex they are on a cycle of length $4$, and $H_2$ has $\gg \delta^2n^2$ edges and every two edges in $H_2$ are contained in a cycle of length at most $8$.","Erdős problems database (status: open; prize: no; tags: graph theory, cycles)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/584,
erdos-585,open,needs-theory,mathematics / graph theory,What is the maximum number of edges that a graph on $n$ vertices can have if it does not contain two edge-disjoint cycles with the same vertex set?,"Erdős problems database (status: open; prize: no; tags: graph theory, cycles)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/585,
erdos-588,open,needs-theory,mathematics / geometry,Let $f_k(n)$ be minimal such that if $n$ points in $\mathbb{R}^2$ have no $k+1$ points on a line then there must be at most $f_k(n)$ many lines containing at least $k$ points. Is it true that\[f_k(n)=o(n^2)\]for $k\geq 4$?,Erdős problems database (status: open; prize: $100; tags: geometry),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A006065, A008997 and check the conjectured growth against them.",https://www.erdosproblems.com/588,
erdos-589,open,needs-theory,mathematics / geometry,Let $g(n)$ be maximal such that in any set of $n$ points in $\mathbb{R}^2$ with no four points on a line there exists a subset on $g(n)$ points with no three points on a line. Estimate $g(n)$.,Erdős problems database (status: open; prize: no; tags: geometry),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/589,
erdos-592,open,needs-theory,mathematics / set theory,"Determine which countable ordinals $\beta$ have the property that, if $\alpha=\omega^{^\beta}$, then in any red/blue colouring of the edges of $K_\alpha$ there is either a red $K_\alpha$ or a blue $K_3$.","Erdős problems database (status: open; prize: $1000; tags: set theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/592,
erdos-593,open,needs-theory,mathematics / set theory,Characterize those finite 3-uniform hypergraphs which appear in every 3-uniform hypergraph of chromatic number $>\aleph_0$.,"Erdős problems database (status: open; prize: $500; tags: set theory, graph theory, hypergraphs, chromatic number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/593,
erdos-595,open,needs-theory,mathematics / graph theory,Is there an infinite graph $G$ which contains no $K_4$ and is not the union of countably many triangle-free graphs?,"Erdős problems database (status: open; prize: $250; tags: graph theory, set theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/595,
erdos-596,open,needs-theory,mathematics / graph theory,"For which graphs $G_1,G_2$ is it true that for every $n\geq 1$ there is a graph $H$ without a $G_1$ but if the edges of $H$ are $n$-coloured then there is a monochromatic copy of $G_2$, and yet for every graph $H$ without a $G_1$ there is an $\aleph_0$-colouring of the edges of $H$ without a monochromatic $G_2$.","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory, set theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/596,
erdos-597,open,needs-theory,mathematics / graph theory,"Let $G$ be a graph on at most $\aleph_1$ vertices which contains no $K_4$ and no $K_{\aleph_0,\aleph_0}$ (the complete bipartite graph with $\aleph_0$ vertices in each class). Is it true that\[\omega_1^2 \to (\omega_1\omega, G)^2?\]What about finite $G$?","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory, set theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/597,
erdos-598,open,needs-theory,mathematics / set theory,Let $m$ be an infinite cardinal and $\kappa$ be the successor cardinal of $2^{\aleph_0}$. Can one colour the countable subsets of $m$ using $\kappa$ many colours so that every $X\subseteq m$ with $\lvert X\rvert=\kappa$ contains subsets of all possible colours?,"Erdős problems database (status: open; prize: no; tags: set theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/598,
erdos-60,open,needs-theory,mathematics / graph theory,Does every graph on $n$ vertices with $>\mathrm{ex}(n;C_4)$ edges contain $\gg n^{1/2}$ many copies of $C_4$?,"Erdős problems database (status: open; prize: no; tags: graph theory, cycles)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A006855 and check the conjectured growth against them.,https://www.erdosproblems.com/60,
erdos-600,open,needs-theory,mathematics / graph theory,"Let $e(n,r)$ be minimal such that every graph on $n$ vertices with at least $e(n,r)$ edges, each edge contained in at least one triangle, must have an edge contained in at least $r$ triangles. Let $r\geq 2$. Is it true that\[e(n,r+1)-e(n,r)\to \infty\]as $n\to \infty$? Is it true that\[\frac{e(n,r+1)}{e(n,r)}\to 1\]as $n\to \infty$?",Erdős problems database (status: open; prize: no; tags: graph theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/600,
erdos-601,open,needs-theory,mathematics / graph theory,For which limit ordinals $\alpha$ is it true that if $G$ is a graph with vertex set $\alpha$ then $G$ must have either an infinite path or independent set on a set of vertices with order type $\alpha$?,"Erdős problems database (status: open; prize: $500; tags: graph theory, set theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/601,
erdos-602,open,needs-theory,mathematics / combinatorics,"Let $(A_i)$ be a family of sets with $\lvert A_i\rvert=\aleph_0$ for all $i$, such that for any $i\neq j$ we have $\lvert A_i\cap A_j\rvert$ finite and $\neq 1$. Is there a $2$-colouring of $\cup A_i$ such that no $A_i$ is monochromatic?","Erdős problems database (status: open; prize: no; tags: combinatorics, set theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/602,
erdos-604,open,needs-theory,mathematics / geometry,"Given $n$ distinct points $A\subset\mathbb{R}^2$ must there be a point $x\in A$ such that\[\#\{ d(x,y) : y \in A\} \gg n^{1-o(1)}?\]Or even $\gg n/\sqrt{\log n}$?","Erdős problems database (status: open; prize: $500; tags: geometry, distances)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/604,
erdos-609,open,needs-theory,mathematics / graph theory,Let $f(n)$ be the minimal $m$ such that if the edges of $K_{2^n+1}$ are coloured with $n$ colours then there must be a monochromatic odd cycle of length at most $m$. Estimate $f(n)$.,"Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/609,
erdos-61,open,needs-theory,mathematics / graph theory,For any graph $H$ is there some $c=c(H)>0$ such that every graph $G$ on $n$ vertices that does not contain $H$ as an induced subgraph contains either a complete graph or independent set on $\geq n^c$ vertices?,Erdős problems database (status: open; prize: no; tags: graph theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/61,
erdos-611,open,needs-theory,mathematics / graph theory,"For a graph $G$ let $\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ (sometimes called the clique transversal number). Is it true that if all maximal cliques in $G$ have at least $cn$ vertices then $\tau(G)=o_c(n)$? Similarly, estimate for $c>0$ the minimal $k_c(n)$ such that if every maximal clique in $G$ has at least $k_c(n)$ vertices then $\tau(G)<(1-c)n$.",Erdős problems database (status: open; prize: no; tags: graph theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/611,
erdos-612,open,needs-theory,mathematics / graph theory,"Let $G$ be a connected graph with $n$ vertices, minimum degree $d$, and diameter $D$. Show if that $G$ contains no $K_{2r}$ and $(r-1)(3r+2)\mid d$ then\[D\leq \frac{2(r-1)(3r+2)}{2r^2-1}\frac{n}{d}+O(1),\]and if $G$ contains no $K_{2r+1}$ and $3r-1 \mid d$ then\[D\leq \frac{3r-1}{r}\frac{n}{d}+O(1).\]",Erdős problems database (status: open; prize: no; tags: graph theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/612,
erdos-614,open,needs-theory,mathematics / graph theory,"Let $f(n,k)$ be minimal such that there is a graph with $n$ vertices and $f(n,k)$ edges where every set of $k+2$ vertices induces a subgraph with maximum degree at least $k$. Determine $f(n,k)$.",Erdős problems database (status: open; prize: no; tags: graph theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/614,
erdos-616,open,needs-theory,mathematics / graph theory,"Let $r\geq 3$. For an $r$-uniform hypergraph $G$ let $\tau(G)$ denote the covering number (or transversal number), the minimum size of a set of vertices which includes at least one from each edge in $G$. Determine the best possible $t$ such that, if $G$ is an $r$-uniform hypergraph $G$ where every subgraph $G'$ on at most $3r-3$ vertices has $\tau(G')\leq 1$, we have $\tau(G)\leq t$.",Erdős problems database (status: open; prize: no; tags: graph theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/616,
erdos-617,open,compute-checkable-small-cases,mathematics / graph theory,"Open, but could be disproved with a finite counterexample. Let $r\geq 3$. If the edges of $K_{r^2+1}$ are $r$-coloured then there exist $r+1$ vertices with at least one colour missing on the edges of the induced $K_{r+1}$.",Erdős problems database (status: falsifiable; prize: no; tags: graph theory),The database marks this as refutable by a finite computation if false: search for a counterexample. Extend OEIS N/A first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/617,
erdos-62,open,needs-theory,mathematics / graph theory,"If $G_1,G_2$ are two graphs with chromatic number $\aleph_1$ then must there exist a graph $G$ whose chromatic number is $4$ (or even $\aleph_0$) which is a subgraph of both $G_1$ and $G_2$?",Erdős problems database (status: open; prize: no; tags: graph theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/62,
erdos-620,open,needs-theory,mathematics / graph theory,If $G$ is a graph on $n$ vertices without a $K_4$ then how large a triangle-free induced subgraph must $G$ contain?,Erdős problems database (status: open; prize: no; tags: graph theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/620,
erdos-623,open,needs-theory,mathematics / set theory,"Let $X$ be a set of cardinality $\aleph_\omega$ and $f$ be a function from the finite subsets of $X$ to $X$ such that $f(A)\not\in A$ for all $A$. Must there exist an infinite $Y\subseteq X$ that is independent - that is, for all finite $B\subset Y$ we have $f(B)\not\in Y$?",Erdős problems database (status: open; prize: no; tags: set theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/623,
erdos-624,open,needs-theory,mathematics / combinatorics,Let $X$ be a finite set of size $n$ and $H(n)$ be such that there is a function $f:\{A : A\subseteq X\}\to X$ so that for every $Y\subseteq X$ with $\lvert Y\rvert \geq H(n)$ we have\[\{ f(A) : A\subseteq Y\}=X.\]Prove that\[H(n)-\log_2 n \to \infty.\],Erdős problems database (status: open; prize: no; tags: combinatorics),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/624,
erdos-625,open,needs-theory,mathematics / graph theory,"The cochromatic number of $G$, denoted by $\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ such that each colour class induces either a complete graph or empty graph. Let $\chi(G)$ denote the chromatic number. If $G$ is a random graph with $n$ vertices and each edge included independently with probability $1/2$ then is it true that almost surely\[\chi(G) - \zeta(G) \to \infty\]as $n\to \infty$?","Erdős problems database (status: open; prize: $1000; tags: graph theory, chromatic number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/625,
erdos-626,open,needs-theory,mathematics / graph theory,"Let $k\geq 4$ and $g_k(n)$ denote the largest $m$ such that there is a graph on $n$ vertices with chromatic number $k$ and girth $>m$ (i.e. contains no cycle of length $\leq m$). Does\[\lim_{n\to \infty}\frac{g_k(n)}{\log n}\]exist? Conversely, if $h^{(m)}(n)$ is the maximal chromatic number of a graph on $n$ vertices with girth $>m$ then does\[\lim_{n\to \infty}\frac{\log h^{(m)}(n)}{\log n}\]exist, and what is its value?","Erdős problems database (status: open; prize: no; tags: graph theory, chromatic number, cycles)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/626,
erdos-627,open,needs-theory,mathematics / graph theory,"Let $\omega(G)$ denote the clique number of $G$ and $\chi(G)$ the chromatic number. If $f(n)$ is the maximum value of $\chi(G)/\omega(G)$, as $G$ ranges over all graphs on $n$ vertices, then does\[\lim_{n\to\infty}\frac{f(n)}{n/(\log_2n)^2}\]exist?","Erdős problems database (status: open; prize: no; tags: graph theory, chromatic number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/627,
erdos-628,open,compute-checkable-small-cases,mathematics / graph theory,"Open, but could be disproved with a finite counterexample. Let $G$ be a graph with chromatic number $k$ containing no $K_k$. If $a,b\geq 2$ and $a+b=k+1$ then must there exist two disjoint subgraphs of $G$ with chromatic numbers $\geq a$ and $\geq b$ respectively?","Erdős problems database (status: falsifiable; prize: no; tags: graph theory, chromatic number)",The database marks this as refutable by a finite computation if false: search for a counterexample. Extend OEIS N/A first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/628,
erdos-629,open,needs-theory,mathematics / graph theory,The list chromatic number $\chi_L(G)$ is defined to be the minimal $k$ such that for any assignment of a list of $k$ colours to each vertex of $G$ (perhaps different lists for different vertices) a colouring of each vertex by a colour on its list can be chosen such that adjacent vertices receive distinct colours. Determine the minimal number of vertices $n(k)$ of a bipartite graph $G$ such that $\chi_L(G)>k$.,"Erdős problems database (status: open; prize: no; tags: graph theory, chromatic number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/629,
erdos-634,open,needs-theory,mathematics / geometry,Find all $n$ such that there is at least one triangle which can be cut into $n$ congruent triangles.,Erdős problems database (status: open; prize: $25; tags: geometry),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/634,
erdos-635,open,needs-theory,mathematics / number theory,"Let $t\geq 1$ and $A\subseteq \{1,\ldots,N\}$ be such that whenever $a,b\in A$ with $b-a\geq t$ we have $b-a\nmid b$. How large can $\lvert A\rvert$ be? Is it true that\[\lvert A\rvert \leq \left(\frac{1}{2}+o_t(1)\right)N?\]",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/635,
erdos-638,open,needs-theory,mathematics / graph theory,Let $S$ be a family of finite graphs such that for every $n$ there is some $G_n\in S$ such that if the edges of $G_n$ are coloured with $n$ colours then there is a monochromatic triangle. Is it true that for every infinite cardinal $\aleph$ there is a graph $G$ of which every finite subgraph is in $S$ and if the edges of $G$ are coloured with $\aleph$ many colours then there is a monochromatic triangle.,"Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/638,
erdos-64,open,compute-checkable-small-cases,mathematics / graph theory,"Open, but could be disproved with a finite counterexample. - $1000 Does every finite graph with minimum degree at least 3 contain a cycle of length $2^k$ for some $k\geq 2$?","Erdős problems database (status: falsifiable; prize: $1000; tags: graph theory, cycles)",The database marks this as refutable by a finite computation if false: search for a counterexample. Extend OEIS N/A first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/64,
erdos-640,open,needs-theory,mathematics / graph theory,Let $k\geq 3$. Does there exist some $f(k)$ such that if a graph $G$ has chromatic number $\geq f(k)$ then $G$ must contain some odd cycle whose vertices span a graph of chromatic number $\geq k$?,"Erdős problems database (status: open; prize: no; tags: graph theory, chromatic number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/640,
erdos-642,open,needs-theory,mathematics / graph theory,Let $f(n)$ be the maximal number of edges in a graph on $n$ vertices such that all cycles have more vertices than chords. Is it true that $f(n)\ll n$?,"Erdős problems database (status: open; prize: no; tags: graph theory, cycles)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/642,
erdos-643,open,needs-theory,mathematics / graph theory,"Let $f(n;t)$ be minimal such that if a $t$-uniform hypergraph on $n$ vertices contains at least $f(n;t)$ edges then there must be four edges $A,B,C,D$ such that\[A\cup B= C\cup D\]and\[A\cap B=C\cap D=\emptyset.\]Estimate $f(n;t)$ - in particular, is it true that for $t\geq 3$\[f(n;t)=(1+o(1))\binom{n}{t-1}?\]","Erdős problems database (status: open; prize: no; tags: graph theory, hypergraphs)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/643,
erdos-644,open,needs-theory,mathematics / combinatorics,"Let $f(k,r)$ be minimal such that if $A_1,A_2,\ldots$ is a family of sets, all of size $k$, such that for every collection of $r$ of the $A_is$ there is some pair $\{x,y\}$ which intersects all of the $A_j$, then there is some set of size $f(k,r)$ which intersects all of the sets $A_i$. Is it true that\[f(k,7)=(1+o(1))\frac{3}{4}k?\]Is it true that for any $r\geq 3$ there exists some constant $c_r$ such that\[f(k,r)=(1+o(1))c_rk?\]",Erdős problems database (status: open; prize: no; tags: combinatorics),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/644,
erdos-647,open,compute-checkable-small-cases,mathematics / number theory,"Open, but could be proved with a finite example. - $44 Let $\tau(n)$ count the number of divisors of $n$. Is there some $n>24$ such that\[\max_{m<n}(m+\tau(m))\leq n+2?\]",Erdős problems database (status: verifiable; prize: £25; tags: number theory),"The database marks this as provable by a finite computation if true: run the verification. Extend OEIS A062249, A087280 first; a new term is a checkable contribution even without a resolution.",https://www.erdosproblems.com/647,
erdos-65,open,needs-theory,mathematics / graph theory,"Let $G$ be a graph with $n$ vertices and $kn$ edges, and $a_1<a_2<\cdots $ be the lengths of cycles in $G$. Is it true that\[\sum\frac{1}{a_i}\gg \log k?\]Is the sum $\sum\frac{1}{a_i}$ minimised when $G$ is a complete bipartite graph?","Erdős problems database (status: open; prize: no; tags: graph theory, cycles)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/65,
erdos-653,open,needs-theory,mathematics / geometry,"Let $x_1,\ldots,x_n\in \mathbb{R}^2$ and let $R(x_i)=\#\{ \lvert x_j-x_i\rvert : j\neq i\}$, where the points are ordered such that\[R(x_1)\leq \cdots \leq R(x_n).\]Let $g(n)$ be the maximum number of distinct values the $R(x_i)$ can take. Is it true that $g(n) \geq (1-o(1))n$?","Erdős problems database (status: open; prize: no; tags: geometry, distances)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/653,
erdos-654,open,needs-theory,mathematics / geometry,"Let $f(n)$ be such that, given any $x_1,\ldots,x_n\in \mathbb{R}^2$ with no four points on a circle, there exists some $x_i$ with at least $f(n)$ many distinct distances to other $x_j$. Estimate $f(n)$ - in particular, is it true that\[f(n)>(1-o(1))n?\]Or at least\[f(n) > (1/3+c)n\]for some $c>0$, for all large $n$?","Erdős problems database (status: open; prize: no; tags: geometry, distances)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/654,
erdos-655,open,needs-theory,mathematics / geometry,"Let $x_1,\ldots,x_n\in \mathbb{R}^2$ be such that no circle whose centre is one of the $x_i$ contains three other points. Are there at least\[(1+c)\frac{n}{2}\]distinct distances determined between the $x_i$, for some constant $c>0$ and all $n$ sufficiently large?","Erdős problems database (status: open; prize: no; tags: geometry, distances)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/655,
erdos-657,open,needs-theory,mathematics / geometry,"Is it true that if $A\subset \mathbb{R}^2$ is a set of $n$ points such that every subset of $3$ points determines $3$ distinct distances (i.e. $A$ has no isosceles triangles) then $A$ must determine at least $f(n)n$ distinct distances, for some $f(n)\to \infty$?","Erdős problems database (status: open; prize: no; tags: geometry, distances)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/657,
erdos-66,open,needs-theory,mathematics / number theory,Is there $A\subseteq \mathbb{N}$ such that\[\lim_{n\to \infty}\frac{1_A\ast 1_A(n)}{\log n}\]exists and is $\neq 0$?,"Erdős problems database (status: open; prize: $500; tags: number theory, additive basis)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/66,
erdos-660,open,needs-theory,mathematics / geometry,"Let $x_1,\ldots,x_n\in \mathbb{R}^3$ be the vertices of a convex polyhedron. Are there at least\[(1-o(1))\frac{n}{2}\]many distinct distances between the $x_i$?","Erdős problems database (status: open; prize: no; tags: geometry, distances, convex)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/660,
erdos-661,open,needs-theory,mathematics / geometry,"Are there, for all large $n$, some points $x_1,\ldots,x_n,y_1,\ldots,y_n\in \mathbb{R}^2$ such that the number of distinct distances $d(x_i,y_j)$ is\[o\left(\frac{n}{\sqrt{\log n}}\right)?\]","Erdős problems database (status: open; prize: $50; tags: geometry, distances)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/661,
erdos-662,open,needs-theory,mathematics / geometry,"Consider the triangular lattice with minimal distance between two points $1$. Denote by $f(t)$ the number of distances from any points $\leq t$. For example $f(1)=6$, $f(\sqrt{3})=12$, and $f(3)=18$. Let $x_1,\ldots,x_n\in \mathbb{R}^2$ be such that $d(x_i,x_j)\geq 1$ for all $i\neq j$. Is it true that, provided $n$ is sufficiently large depending on $t$, the number of distances $d(x_i,x_j)\leq t$ is less than or equal to $f(t)$ with equality perhaps only for the triangular lattice? In particular, is it true that the number of distances $\leq \sqrt{3}-\epsilon$ is less than $1$?","Erdős problems database (status: open; prize: no; tags: geometry, distances)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/662,
erdos-663,open,needs-theory,mathematics / number theory,"Let $k\geq 2$ and $q(n,k)$ denote the least prime which does not divide $\prod_{1\leq i\leq k}(n+i)$. Is it true that, if $k$ is fixed and $n$ is sufficiently large, we have\[q(n,k)<(1+o(1))\log n?\]",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A391668 and check the conjectured growth against them.,https://www.erdosproblems.com/663,
erdos-665,open,needs-theory,mathematics / combinatorics,"A pairwise balanced design for $\{1,\ldots,n\}$ is a collection of sets $A_1,\ldots,A_m\subseteq \{1,\ldots,n\}$ such that $2\leq \lvert A_i\rvert <n$ and every pair of distinct elements $x,y\in \{1,\ldots,n\}$ is contained in exactly one $A_i$. Is there a constant $C>0$ and, for all large $n$, a pairwise balanced design such that\[\lvert A_i\rvert > n^{1/2}-C\]for all $1\leq i\leq m$?",Erdős problems database (status: open; prize: no; tags: combinatorics),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/665,
erdos-667,open,needs-theory,mathematics / graph theory,"Let $p,q\geq 1$ be fixed integers. We define $H(n)=H(N;p,q)$ to be the largest $m$ such that any graph on $n$ vertices where every set of $p$ vertices spans at least $q$ edges must contain a complete graph on $m$ vertices. Is\[c(p,q)=\liminf \frac{\log H(n)}{\log n}\]a strictly increasing function of $q$ for $1\leq q\leq \binom{p-1}{2}+1$?","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/667,
erdos-668,open,needs-theory,mathematics / geometry,Is it true that the number of incongruent sets of $n$ points in $\mathbb{R}^2$ which maximise the number of unit distances tends to infinity as $n\to\infty$? Is it always $>1$ for $n>3$?,"Erdős problems database (status: open; prize: no; tags: geometry, distances)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A385657 and check the conjectured growth against them.,https://www.erdosproblems.com/668,
erdos-669,open,needs-theory,mathematics / geometry,"Let $F_k(n)$ be minimal such that for any $n$ points in $\mathbb{R}^2$ there exist at most $F_k(n)$ many distinct lines passing through at least $k$ of the points, and $f_k(n)$ similarly but with lines passing through exactly $k$ points. Estimate $f_k(n)$ and $F_k(n)$ - in particular, determine $\lim F_k(n)/n^2$ and $\lim f_k(n)/n^2$.",Erdős problems database (status: open; prize: no; tags: geometry),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A003035, A006065, A008997, possible and check the conjectured growth against them.",https://www.erdosproblems.com/669,
erdos-670,open,needs-theory,mathematics / geometry,Let $A\subseteq \mathbb{R}^d$ be a set of $n$ points such that all pairwise distances differ by at least $1$. Is the diameter of $A$ at least $(1+o(1))n^2$?,"Erdős problems database (status: open; prize: no; tags: geometry, distances)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/670,
erdos-671,open,needs-theory,mathematics / analysis,"Given $a_{i}^n\in [-1,1]$ for all $1\leq i\leq n<\infty$ we define $p_{i}^n$ as the unique polynomial of degree $n-1$ such that $p_{i}^n(a_{i}^n)=1$ and $p_{i}^n(a_{i'}^n)=0$ if $1\leq i'\leq n$ with $i\neq i'$. We similarly define\[\mathcal{L}^nf(x) = \sum_{1\leq i\leq n}f(a_i^n)p_i^n(x),\]the unique polynomial of degree $n-1$ which agrees with $f$ on $a_i^n$ for $1\leq i\leq n$ (that is, the sequence of Lagrange interpolation polynomials). Is there such a sequence of $a_i^n$ such that for every continuous $f:[-1,1]\to \mathbb{R}$ there exists some $x\in [-1,1]$ where\[\limsup_{n\to \infty} \sum_{1\leq i\leq n}\lvert p_{i}^n(x)\rvert=\infty\]and yet\[\mathcal{L}^nf(x) \to f(x)?\]Is there such a sequence such that\[\limsup_{n\to \infty} \sum_{1\leq i\leq n}\lvert p_{i}^n(x)\rvert=\infty\]for every $x\in [-1,1]$ and yet for every continuous $f:[-1,1]\to \mathbb{R}$ there exists $x\in [-1,",Erdős problems database (status: open; prize: $250; tags: analysis),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/671,
erdos-672,open,compute-checkable-small-cases,mathematics / number theory,"Open, but could be proved with a finite example. Can the product of an arithmetic progression of positive integers $n,n+d,\ldots,n+(k-1)d$ of length $k\geq 4$ (with $(n,d)=1$) be a perfect power?",Erdős problems database (status: verifiable; prize: no; tags: number theory),The database marks this as provable by a finite computation if true: run the verification. Extend OEIS N/A first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/672,
erdos-675,open,needs-theory,mathematics / number theory,"We say that $A\subset \mathbb{N}$ has the translation property if, for every $n$, there exists some integer $t_n\geq 1$ such that, for all $1\leq a\leq n$,\[a\in A\quad\textrm{ if and only if }\quad a+t_n\in A.\] Does the set of the sums of two squares have the translation property? If we partition all primes into $P\sqcup Q$, such that each set contains $\gg x/\log x$ many primes $\leq x$ for all large $x$, then can the set of integers only divisible by primes from $P$ have the translation property? If $A$ is the set of squarefree numbers then how fast does the minimal such $t_n$ grow? Is it true that $t_n>\exp(n^c)$ for some constant $c>0$?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/675,
erdos-676,open,needs-theory,mathematics / number theory,Is every sufficiently large integer of the form\[ap^2+b\]for some prime $p$ and integer $a\geq 1$ and $0\leq b<p$?,Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A390181, in progress and check the conjectured growth against them.",https://www.erdosproblems.com/676,
erdos-677,open,needs-theory,mathematics / number theory,"Let $M(n,k)=[n+1,\ldots,n+k]$ be the least common multiple of $\{n+1,\ldots,n+k\}$. Is it true that for all $m\geq n+k$\[M(n,k) \neq M(m,k)?\]",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/677,
erdos-679,open,needs-theory,mathematics / number theory,Let $\epsilon>0$ and $\omega(n)$ count the number of distinct prime factors of $n$. Are there infinitely many values of $n$ such that\[\omega(n-k) < (1+\epsilon)\frac{\log k}{\log\log k}\]for all $k<n$ which are sufficiently large depending on $\epsilon$ only? Can one show the stronger version with\[\omega(n-k) < \frac{\log k}{\log\log k}+O(1)\]is false?,Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/679,
erdos-68,open,needs-theory,mathematics / number theory,Is\[\sum_{n\geq 2}\frac{1}{n!-1}\]irrational?,"Erdős problems database (status: open; prize: no; tags: number theory, irrationality)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A331373 and check the conjectured growth against them.,https://www.erdosproblems.com/68,
erdos-680,open,needs-theory,mathematics / number theory,"Is it true that, for all sufficiently large $n$, there exists some $k$ such that\[p(n+k)>k^2+1,\]where $p(m)$ denotes the least prime factor of $m$? Can one prove this is false if we replace $k^2+1$ by $e^{(1+\epsilon)\sqrt{k}}+C_\epsilon$, for all $\epsilon>0$, where $C_\epsilon>0$ is some constant?","Erdős problems database (status: open; prize: no; tags: number theory, primes)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/680,
erdos-681,open,needs-theory,mathematics / number theory,"Is it true that for all large $n$ there exists $k$ such that $n+k$ is composite and\[p(n+k)>k^2,\]where $p(m)$ is the least prime factor of $m$?","Erdős problems database (status: open; prize: no; tags: number theory, primes)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A389680 and check the conjectured growth against them.,https://www.erdosproblems.com/681,
erdos-683,open,needs-theory,mathematics / number theory,"Is it true that for every $1\leq k\leq n$ the largest prime divisor of $\binom{n}{k}$, say $P(\binom{n}{k})$, satisfies\[P\left(\binom{n}{k}\right)\geq \min(n-k+1, k^{1+c})\]for some constant $c>0$?","Erdős problems database (status: open; prize: no; tags: number theory, primes, binomial coefficients)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A006530, A074399, A121359, possible and check the conjectured growth against them.",https://www.erdosproblems.com/683,
erdos-684,open,needs-theory,mathematics / number theory,"For $0\leq k\leq n$ write\[\binom{n}{k} = uv\]where the only primes dividing $u$ are in $[2,k]$ and the only primes dividing $v$ are in $(k,n]$. Let $f(n)$ be the smallest $k$ such that $u>n^2$. Give bounds for $f(n)$.","Erdős problems database (status: open; prize: no; tags: number theory, primes, binomial coefficients)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A392019, possible and check the conjectured growth against them.",https://www.erdosproblems.com/684,
erdos-685,open,needs-theory,mathematics / number theory,Let $\epsilon>0$ and $n$ be large depending on $\epsilon$. Is it true that for all $n^\epsilon<k\leq n^{1-\epsilon}$ the number of distinct prime divisors of $\binom{n}{k}$ is\[(1+o(1))k\sum_{k<p<n}\frac{1}{p}?\]Or perhaps even when $k \geq (\log n)^c$?,"Erdős problems database (status: open; prize: no; tags: number theory, primes, binomial coefficients)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/685,
erdos-686,open,needs-theory,mathematics / number theory,Can every integer $N\geq 2$ be written as\[N=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)}\]for some $k\geq 2$ and $m\geq n+k$?,Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/686,
erdos-687,open,needs-theory,mathematics / number theory,"Let $Y(x)$ be the maximal $y$ such that there exists a choice of congruence classes $a_p$ for all primes $p\leq x$ such that every integer in $[1,y]$ is congruent to at least one of the $a_p\pmod{p}$. Give good estimates for $Y(x)$. In particular, can one prove that $Y(x)=o(x^2)$ or even $Y(x)\ll x^{1+o(1)}$?",Erdős problems database (status: open; prize: $1000; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A048670, A058989 and check the conjectured growth against them.",https://www.erdosproblems.com/687,
erdos-688,open,needs-theory,mathematics / number theory,"Define $\epsilon_n$ to be maximal such that there exists some choice of congruence class $a_p$ for all primes $n^{\epsilon_n}<p\leq n$ such that every integer in $[1,n]$ satisfies at least one of the congruences $\equiv a_p\pmod{p}$. Estimate $\epsilon_n$ - in particular is it true that $\epsilon_n=o(1)$?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/688,
erdos-689,open,needs-theory,mathematics / number theory,"Let $n$ be sufficiently large. Is there some choice of congruence class $a_p$ for all primes $2\leq p\leq n$ such that every integer in $[1,n]$ satisfies at least two of the congruences $\equiv a_p\pmod{p}$?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/689,
erdos-691,open,needs-theory,mathematics / number theory,Given $A\subseteq \mathbb{N}$ let $M_A=\{ n \geq 1 : a\mid n\textrm{ for some }a\in A\}$ be the set of multiples of $A$. Find a necessary and sufficient condition on $A$ for $M_A$ to have density $1$.,Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/691,
erdos-693,open,needs-theory,mathematics / number theory,"Let $k\geq 2$ and $n$ be sufficiently large depending on $k$. Let $A=\{a_1<a_2<\cdots \}$ be the set of those integers in $[n,n^k]$ which have a divisor in $(n,2n)$. Estimate\[\max_{i} a_{i+1}-a_i.\]Is this $\leq (\log n)^{O(1)}$?","Erdős problems database (status: open; prize: no; tags: number theory, divisors)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A391118, possible and check the conjectured growth against them.",https://www.erdosproblems.com/693,
erdos-695,open,needs-theory,mathematics / number theory,Let $p_1<p_2<\cdots$ be a sequence of primes such that $p_{i+1}\equiv 1\pmod{p_i}$. Is it true that\[\lim_k p_k^{1/k}=\infty?\]Does there exist such a sequence with\[p_k \leq \exp(k(\log k)^{1+o(1)})?\],Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A061092 and check the conjectured growth against them.,https://www.erdosproblems.com/695,
erdos-699,open,compute-checkable-small-cases,mathematics / number theory,"Open, but could be disproved with a finite counterexample. Is it true that for every $1\leq i<j\leq n/2$ there exists some prime $p\geq i$ such that\[p\mid \textrm{gcd}\left(\binom{n}{i}, \binom{n}{j}\right)?\]","Erdős problems database (status: falsifiable; prize: no; tags: number theory, binomial coefficients)",The database marks this as refutable by a finite computation if false: search for a counterexample. Extend OEIS N/A first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/699,
erdos-7,open,compute-checkable-small-cases,mathematics / number theory,"Open, but could be proved with a finite example. - $25 Is there a distinct covering system all of whose moduli are odd?","Erdős problems database (status: verifiable; prize: no; tags: number theory, covering systems)",The database marks this as provable by a finite computation if true: run the verification. Extend OEIS N/A first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/7,
erdos-70,open,needs-theory,mathematics / graph theory,"Let $\mathfrak{c}$ be the ordinal of the real numbers, $\beta$ be any countable ordinal, and $2\leq n<\omega$. Is it true that $\mathfrak{c}\to (\beta, n)_2^3$?","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory, set theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/70,
erdos-700,open,needs-theory,mathematics / number theory,"Let\[f(n)=\min_{1<k\leq n/2}\textrm{gcd}\left(n,\binom{n}{k}\right).\] Characterise those composite $n$ such that $f(n)=n/P(n)$, where $P(n)$ is the largest prime dividing $n$. Are there infinitely many composite $n$ such that $f(n)>n^{1/2}$? Is it true that, for every composite $n$,\[f(n) \ll_A \frac{n}{(\log n)^A}\]for every $A>0$?","Erdős problems database (status: open; prize: no; tags: number theory, binomial coefficients)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A091963, possible and check the conjectured growth against them.",https://www.erdosproblems.com/700,
erdos-701,open,needs-theory,mathematics / combinatorics,Let $\mathcal{F}$ be a family of sets closed under taking subsets (i.e. if $B\subseteq A\in\mathcal{F}$ then $B\in \mathcal{F}$). There exists some element $x$ such that whenever $\mathcal{F}'\subseteq \mathcal{F}$ is an intersecting subfamily we have\[\lvert \mathcal{F}'\rvert \leq \lvert \{ A\in \mathcal{F} : x\in A\}\rvert.\],"Erdős problems database (status: open; prize: no; tags: combinatorics, intersecting family)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/701,
erdos-704,open,needs-theory,mathematics / graph theory,"Let $G_n$ be the unit distance graph in $\mathbb{R}^n$, with two vertices joined by an edge if and only if the distance between them is $1$. Estimate the chromatic number $\chi(G_n)$. Does it grow exponentially in $n$? Does\[\lim_{n\to \infty}\chi(G_n)^{1/n}\]exist?","Erdős problems database (status: open; prize: no; tags: graph theory, geometry, chromatic number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/704,
erdos-706,open,needs-theory,mathematics / graph theory,"Let $L(r)$ be such that if $G$ is a graph formed by taking a finite set of points $P$ in $\mathbb{R}^2$ and some set $A\subset (0,\infty)$ of size $r$, where the vertex set is $P$ and there is an edge between two points if and only if their distance is a member of $A$, then $\chi(G)\leq L(r)$. Estimate $L(r)$. In particular, is it true that $L(r)\leq r^{O(1)}$?","Erdős problems database (status: open; prize: no; tags: graph theory, chromatic number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/706,
erdos-708,open,needs-theory,mathematics / number theory,"Let $g(n)$ be minimal such that for any $A\subseteq [2,\infty)\cap \mathbb{N}$ with $\lvert A\rvert =n$ and any set $I$ of $\max(A)$ consecutive integers there exists some $B\subseteq I$ with $\lvert B\rvert=g(n)$ such that\[\prod_{a\in A} a \mid \prod_{b\in B}b.\]Is it true that\[g(n) \leq (2+o(1))n?\]Or perhaps even $g(n)\leq 2n$?",Erdős problems database (status: open; prize: $100; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/708,
erdos-709,open,needs-theory,mathematics / number theory,"Let $f(n)$ be minimal such that, for any $A=\{a_1,\ldots,a_n\}\subseteq [2,\infty)\cap\mathbb{N}$ of size $n$, in any interval $I$ of $f(n)\max(A)$ consecutive integers there exist distinct $x_1,\ldots,x_n\in I$ such that $a_i\mid x_i$. Obtain good bounds for $f(n)$, or even an asymptotic formula.",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/709,
erdos-710,open,needs-theory,mathematics / number theory,"Let $f(n)$ be minimal such that in $(n,n+f(n))$ there exist distinct integers $a_1,\ldots,a_n$ such that $k\mid a_k$ for all $1\leq k\leq n$. Obtain an asymptotic formula for $f(n)$.",Erdős problems database (status: open; prize: ₹2000; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A390246 and check the conjectured growth against them.,https://www.erdosproblems.com/710,
erdos-711,open,needs-theory,mathematics / number theory,"Let $f(n,m)$ be minimal such that in $(m,m+f(n,m))$ there exist distinct integers $a_1,\ldots,a_n$ such that $k\mid a_k$ for all $1\leq k\leq n$. Prove that\[\max_m f(n,m) \leq n^{1+o(1)}\]and that\[\max_m (f(n,m)-f(n,n))\to \infty.\]",Erdős problems database (status: open; prize: ₹1000; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/711,
erdos-712,open,needs-theory,mathematics / graph theory,"Determine, for any $k>r>2$, the value of\[\frac{\mathrm{ex}_r(n,K_k^r)}{\binom{n}{r}},\]where $\mathrm{ex}_r(n,K_k^r)$ is the largest number of $r$-edges which can placed on $n$ vertices so that there exists no set of $k$ vertices which is covered by all $\binom{k}{r}$ possible $r$-edges.","Erdős problems database (status: open; prize: $500; tags: graph theory, turan number, hypergraphs)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/712,
erdos-713,open,needs-theory,mathematics / graph theory,"Is it true that, for every bipartite graph $G$, there exists some $\alpha\in [1,2)$ and $c>0$ such that\[\mathrm{ex}(n;G)\sim cn^\alpha?\]Must $\alpha$ be rational?","Erdős problems database (status: open; prize: $500; tags: graph theory, turan number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/713,
erdos-714,open,needs-theory,mathematics / graph theory,"Is it true that\[\mathrm{ex}(n; K_{r,r}) \gg n^{2-1/r}?\]","Erdős problems database (status: open; prize: no; tags: graph theory, turan number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/714,
erdos-719,open,needs-theory,mathematics / graph theory,"Let $\mathrm{ex}_r(n;K_{r+1}^r)$ be the maximum number of $r$-edges that can be placed on $n$ vertices without forming a $K_{r+1}^r$ (the $r$-uniform complete graph on $r+1$ vertices). Is every $r$-hypergraph $G$ on $n$ vertices the union of at most $\mathrm{ex}_{r}(n;K_{r+1}^r)$ many copies of $K_r^r$ and $K_{r+1}^r$, no two of which share a $K_r^r$?","Erdős problems database (status: open; prize: no; tags: graph theory, hypergraphs)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/719,
erdos-723,open,compute-checkable-small-cases,mathematics / combinatorics,"Open, but could be disproved with a finite counterexample. If there is a finite projective plane of order $n$ then must $n$ be a prime power? A finite projective plane of order $n$ is a collection of subsets of $\{1,\ldots,n^2+n+1\}$ of size $n+1$ such that every pair of elements is contained in exactly one set.",Erdős problems database (status: falsifiable; prize: no; tags: combinatorics),The database marks this as refutable by a finite computation if false: search for a counterexample. Extend OEIS N/A first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/723,
erdos-724,open,needs-theory,mathematics / combinatorics,Let $f(n)$ be the maximum number of mutually orthogonal Latin squares of order $n$. Is it true that\[f(n) \gg n^{1/2}?\],Erdős problems database (status: open; prize: no; tags: combinatorics),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A001438 and check the conjectured growth against them.,https://www.erdosproblems.com/724,
erdos-725,open,needs-theory,mathematics / combinatorics,Give an asymptotic formula for the number of $k\times n$ Latin rectangles .,Erdős problems database (status: open; prize: no; tags: combinatorics),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A001009 and check the conjectured growth against them.,https://www.erdosproblems.com/725,
erdos-726,open,needs-theory,mathematics / number theory,"As $n\to \infty$ ranges over integers\[\sum_{p\leq n}1_{n\in (p/2,p)\pmod{p}}\frac{1}{p}\sim \frac{\log\log n}{2}.\]",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/726,
erdos-727,open,needs-theory,mathematics / number theory,Let $k\geq 2$. Does\[(n+k)!^2 \mid (2n)!\]for infinitely many $n$?,"Erdős problems database (status: open; prize: no; tags: number theory, factorials)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A002503, A343507, A389396 and check the conjectured growth against them.",https://www.erdosproblems.com/727,
erdos-731,open,needs-theory,mathematics / number theory,"Find some reasonable function $f(n)$ such that, for almost all integers $n$, the least integer $m$ such that $m\nmid \binom{2n}{n}$ satisfies\[m\sim f(n).\]","Erdős problems database (status: open; prize: no; tags: number theory, binomial coefficients)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A006197 and check the conjectured growth against them.,https://www.erdosproblems.com/731,
erdos-734,open,needs-theory,mathematics / combinatorics,"Find, for all large $n$, a non-trivial pairwise balanced block design $A_1,\ldots,A_m\subseteq \{1,\ldots,n\}$ such that, for all $t$, there are $O(n^{1/2})$ many $i$ such that $\lvert A_i\rvert=t$.",Erdős problems database (status: open; prize: no; tags: combinatorics),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/734,
erdos-736,open,needs-theory,mathematics / graph theory,"Open in general, but there exist models of set theory where the result is false. Let $G$ be a graph with chromatic number $\aleph_1$. Is there, for every cardinal number $m$, some graph $G_m$ of chromatic number $m$ such that every finite subgraph of $G_m$ is a subgraph of $G$?","Erdős problems database (status: not provable; prize: no; tags: graph theory, chromatic number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/736,
erdos-738,open,needs-theory,mathematics / graph theory,If $G$ has infinite chromatic number and is triangle-free (contains no $K_3$) then must $G$ contain every tree as an induced subgraph?,"Erdős problems database (status: open; prize: no; tags: graph theory, chromatic number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/738,
erdos-739,open,needs-theory,mathematics / graph theory,"Open in general, but there exist models of set theory where the result is false. Let $\mathfrak{m}$ be an infinite cardinal and $G$ be a graph with chromatic number $\mathfrak{m}$. Is it true that, for every infinite cardinal $\mathfrak{n}< \mathfrak{m}$, there exists a subgraph of $G$ with chromatic number $\mathfrak{n}$?","Erdős problems database (status: not provable; prize: no; tags: graph theory, chromatic number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/739,
erdos-74,open,needs-theory,mathematics / graph theory,Let $f(n)\to \infty$ (possibly very slowly). Is there a graph of infinite chromatic number such that every finite subgraph on $n$ vertices can be made bipartite by deleting at most $f(n)$ edges?,"Erdős problems database (status: open; prize: $500; tags: graph theory, chromatic number, cycles)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/74,
erdos-740,open,needs-theory,mathematics / graph theory,Let $\mathfrak{m}$ be an infinite cardinal and $G$ be a graph with chromatic number $\mathfrak{m}$. Let $r\geq 1$. Must $G$ contain a subgraph of chromatic number $\mathfrak{m}$ which does not contain any odd cycle of length $\leq r$?,"Erdős problems database (status: open; prize: no; tags: graph theory, chromatic number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/740,
erdos-742,open,compute-checkable-small-cases,mathematics / graph theory,"Resolved up to a finite check. Let $G$ be a graph on $n$ vertices with diameter $2$, such that deleting any edge increases the diameter of $G$. Is it true that $G$ has at most $n^2/4$ edges?",Erdős problems database (status: decidable; prize: no; tags: graph theory),The database marks this as reduced to a finite computation: run it. Extend OEIS N/A first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/742,
erdos-743,open,compute-checkable-small-cases,mathematics / graph theory,"Open, but could be disproved with a finite counterexample. Let $T_2,\ldots,T_n$ be a collection of trees such that $T_k$ has $k$ vertices. Can we always write $K_n$ as the edge disjoint union of the $T_k$?",Erdős problems database (status: falsifiable; prize: no; tags: graph theory),The database marks this as refutable by a finite computation if false: search for a counterexample. Extend OEIS N/A first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/743,
erdos-749,open,needs-theory,mathematics / additive combinatorics,Let $\epsilon>0$. Does there exist $A\subseteq \mathbb{N}$ such that the lower density of $A+A$ is at least $1-\epsilon$ and yet $1_A\ast 1_A(n) \ll_\epsilon 1$ for all $n$?,Erdős problems database (status: open; prize: no; tags: additive combinatorics),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/749,
erdos-75,open,needs-theory,mathematics / graph theory,Is there a graph of chromatic number $\aleph_1$ with $\aleph_1$ vertices such that for all $\epsilon>0$ if $n$ is sufficiently large and $H$ is a subgraph on $n$ vertices then $H$ contains an independent set of size $>n^{1-\epsilon}$? What about an independent set of size $\gg n$?,"Erdős problems database (status: open; prize: no; tags: graph theory, chromatic number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/75,
erdos-757,open,needs-theory,mathematics / geometry,Let $A\subset \mathbb{R}$ be a set of size $n$ such that every subset $B\subseteq A$ with $\lvert B\rvert =4$ has $\lvert B-B\rvert\geq 11$. Find the best constant $c>0$ such that $A$ must always contain a Sidon set of size $\geq cn$.,"Erdős problems database (status: open; prize: no; tags: geometry, distances, sidon sets)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/757,
erdos-761,open,needs-theory,mathematics / graph theory,"The cochromatic number of $G$, denoted by $\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ such that each colour class induces either a complete graph or empty graph. The dichromatic number of $G$, denoted by $\delta(G)$, is the minimum number $k$ of colours required such that, in any orientation of the edges of $G$, there is a $k$-colouring of the vertices of $G$ such that there are no monochromatic oriented cycles. Must a graph with large chromatic number have large dichromatic number? Must a graph with large cochromatic number contain a graph with large dichromatic number?","Erdős problems database (status: open; prize: no; tags: graph theory, chromatic number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/761,
erdos-766,open,needs-theory,mathematics / graph theory,"Let $f(n;k,l)=\min \mathrm{ex}(n;G)$, where $G$ ranges over all graphs with $k$ vertices and $l$ edges. Give good estimates for $f(n;k,l)$ in the range $k<l\leq k^2/4$. For fixed $k$ and large $n$ is $f(n;k,l)$ a strictly monotone function of $l$?","Erdős problems database (status: open; prize: no; tags: graph theory, turan number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/766,
erdos-768,open,needs-theory,mathematics / number theory,"Let $A\subset\mathbb{N}$ be the set of $n$ such that for every prime $p\mid n$ there exists some $d\mid n$ with $d>1$ such that $d\equiv 1\pmod{p}$. Is it true that there exists some constant $c>0$ such that for all large $N$\[\frac{\lvert A\cap [1,N]\rvert}{N}=\exp(-(c+o(1))\sqrt{\log N}\log\log N).\]",Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A001034, A352287 and check the conjectured growth against them.",https://www.erdosproblems.com/768,
erdos-769,open,needs-theory,mathematics / number theory,"Let $c(n)$ be minimal such that if $k\geq c(n)$ then the $n$-dimensional unit cube can be decomposed into $k$ homothetic $n$-dimensional cubes. Give good bounds for $c(n)$ - in particular, is it true that $c(n) \gg n^n$?","Erdős problems database (status: open; prize: no; tags: number theory, geometry)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A014544, possible and check the conjectured growth against them.",https://www.erdosproblems.com/769,
erdos-77,open,needs-theory,mathematics / graph theory,"If $R(k)$ is the Ramsey number for $K_k$, the minimal $n$ such that every $2$-colouring of the edges of $K_n$ contains a monochromatic copy of $K_k$, then find the value of\[\lim_{k\to \infty}R(k)^{1/k}.\]","Erdős problems database (status: open; prize: $250; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A059442 and check the conjectured growth against them.,https://www.erdosproblems.com/77,
erdos-770,open,needs-theory,mathematics / number theory,"Let $h(n)$ be minimal such that $2^n-1,3^n-1,\ldots,h(n)^n-1$ are mutually coprime. Does, for every prime $p$, the density $\delta_p$ of integers with $h(n)=p$ exist? Does $\liminf h(n)=\infty$? Is it true that if $p$ is the greatest prime such that $p-1\mid n$ and $p>n^\epsilon$ then $h(n)=p$?",Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A263647, possible and check the conjectured growth against them.",https://www.erdosproblems.com/770,
erdos-773,open,needs-theory,mathematics / number theory,"What is the size of the largest Sidon subset $A\subseteq\{1,2^2,\ldots,N^2\}$? Is it $N^{1-o(1)}$?","Erdős problems database (status: open; prize: no; tags: number theory, sidon sets, squares)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A390813 and check the conjectured growth against them.,https://www.erdosproblems.com/773,
erdos-774,open,needs-theory,mathematics / number theory,"We call $A\subset \mathbb{N}$ dissociated if $\sum_{n\in X}n\neq \sum_{m\in Y}m$ for all finite $X,Y\subset A$ with $X\neq Y$. Let $A\subset \mathbb{N}$ be an infinite set. We call $A$ proportionately dissociated if every finite $B\subset A$ contains a dissociated set of size $\gg \lvert B\rvert$. Is every proportionately dissociated set the union of a finite number of dissociated sets?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/774,
erdos-776,open,needs-theory,mathematics / combinatorics,"Let $r\geq 2$ and $A_1,\ldots,A_m\subseteq \{1,\ldots,n\}$ be such that $A_i\not\subseteq A_j$ for all $i\neq j$ and for any $t$ if there exists some $i$ with $\lvert A_i\rvert=t$ then there must exist at least $r$ sets of that size. How large must $n$ be (as a function of $r$) to ensure that there is such a family which achieves $n-3$ distinct sizes of sets?",Erdős problems database (status: open; prize: no; tags: combinatorics),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/776,
erdos-778,open,needs-theory,mathematics / graph theory,"Alice and Bob play a game on the edges of $K_n$, alternating colouring edges by red (Alice) and blue (Bob). Alice goes first, and wins if at the end the largest red clique is larger than any of the blue cliques. Does Bob have a winning strategy for $n\geq 3$? (Erdős believed the answer is yes.) If we change the game so that Bob colours two edges after each edge that Alice colours, but now require Bob's largest clique to be strictly larger than Alice's, then does Bob have a winning strategy for $n>3$? Finally, consider the game when Alice wins if the maximum degree of the red subgraph is larger than the maximum degree of the blue subgraph. Who wins?",Erdős problems database (status: open; prize: no; tags: graph theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/778,
erdos-779,open,compute-checkable-small-cases,mathematics / number theory,"Open, but could be disproved with a finite counterexample. Let $n> 1$ and $p_1<\cdots<p_n$ denote the first $n$ primes. Let $P=\prod_{1\leq i\leq n}p_i$. Does there always exist some prime $p$ with $p_n<p<P$ such that $P+p$ is prime?","Erdős problems database (status: falsifiable; prize: no; tags: number theory, primes)",The database marks this as refutable by a finite computation if false: search for a counterexample. Extend OEIS A005235 first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/779,
erdos-78,open,needs-theory,mathematics / graph theory,"Let $R(k)$ be the Ramsey number for $K_k$, the minimal $n$ such that every $2$-colouring of the edges of $K_n$ contains a monochromatic copy of $K_k$. Give a constructive proof that $R(k)>C^k$ for some constant $C>1$.","Erdős problems database (status: open; prize: $100; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A059442 and check the conjectured growth against them.,https://www.erdosproblems.com/78,
erdos-782,open,needs-theory,mathematics / number theory,"Do the squares contain arbitrarily long quasi-progressions? That is, does there exist some constant $C>0$ such that, for any $k$, the squares contain a sequence $x_1,\ldots,x_k$ where, for some $d$ and all $1\leq i<k$,\[x_i+d\leq x_{i+1}\leq x_i+d+C.\]Do the squares contain arbitrarily large cubes\[a+\left\{ \sum_i \epsilon_ib_i : \epsilon_i\in \{0,1\}\right\}?\]",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/782,
erdos-786,open,needs-theory,mathematics / number theory,"Let $\epsilon>0$. Is there some set $A\subset \mathbb{N}$ of density $>1-\epsilon$ such that $a_1\cdots a_r=b_1\cdots b_s$ with $a_i,b_j\in A$ can only hold when $r=s$? Similarly, can one always find a set $A\subset\{1,\ldots,N\}$ with this property of size $\geq (1-o(1))N$?",Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A143301, possible and check the conjectured growth against them.",https://www.erdosproblems.com/786,
erdos-787,open,needs-theory,mathematics / additive combinatorics,Let $g(n)$ be maximal such that given any set $A\subset \mathbb{R}$ with $\lvert A\rvert=n$ there exists some $B\subseteq A$ of size $\lvert B\rvert\geq g(n)$ such that $b_1+b_2\not\in A$ for all $b_1\neq b_2\in B$. Estimate $g(n)$.,Erdős problems database (status: open; prize: no; tags: additive combinatorics),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/787,
erdos-788,open,needs-theory,mathematics / additive combinatorics,"Let $f(n)$ be maximal such that if $B\subset (2n,4n)\cap \mathbb{N}$ there exists some $C\subset (n,2n)\cap \mathbb{N}$ such that $c_1+c_2\not\in B$ for all $c_1\neq c_2\in C$ and $\lvert C\rvert+\lvert B\rvert \geq f(n)$. Estimate $f(n)$. In particular is it true that $f(n)\leq n^{1/2+o(1)}$?",Erdős problems database (status: open; prize: no; tags: additive combinatorics),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/788,
erdos-789,open,needs-theory,mathematics / additive combinatorics,"Let $h(n)$ be maximal such that if $A\subseteq \mathbb{Z}$ with $\lvert A\rvert=n$ then there is $B\subseteq A$ with $\lvert B\rvert \geq h(n)$ such that if $a_1+\cdots+a_r=b_1+\cdots+b_s$ with $a_i,b_i\in B$ then $r=s$. Estimate $h(n)$.",Erdős problems database (status: open; prize: no; tags: additive combinatorics),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/789,
erdos-790,open,needs-theory,mathematics / additive combinatorics,"Let $l(n)$ be maximal such that if $A\subset\mathbb{Z}$ with $\lvert A\rvert=n$ then there exists a sum-free $B\subseteq A$ with $\lvert B\rvert \geq l(n)$ - that is, $B$ is such that there are no solutions to\[a_1=a_2+\cdots+a_r\]with $a_i\in B$ all distinct. Estimate $l(n)$. In particular, is it true that $l(n)n^{-1/2}\to \infty$? Is it true that $l(n)< n^{1-c}$ for some $c>0$?",Erdős problems database (status: open; prize: no; tags: additive combinatorics),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/790,
erdos-791,open,needs-theory,mathematics / additive combinatorics,"Let $g(n)$ be minimal such that there exists $A\subseteq \{0,\ldots,n\}$ of size $g(n)$ with $\{0,\ldots,n\}\subseteq A+A$. Estimate $g(n)$. In particular is it true that $g(n)\sim 2n^{1/2}$?",Erdős problems database (status: open; prize: no; tags: additive combinatorics),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A066063 and check the conjectured growth against them.,https://www.erdosproblems.com/791,
erdos-792,open,needs-theory,mathematics / additive combinatorics,"Let $f(n)$ be maximal such that in any $A\subset \mathbb{Z}$ with $\lvert A\rvert=n$ there exists some sum-free subset $B\subseteq A$ with $\lvert B\rvert \geq f(n)$, so that there are no solutions to\[a+b=c\]with $a,b,c\in B$. Estimate $f(n)$.",Erdős problems database (status: open; prize: no; tags: additive combinatorics),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/792,
erdos-796,open,needs-theory,mathematics / number theory,"Let $k\geq 2$ and let $g_k(n)$ be the largest possible size of $A\subseteq \{1,\ldots,n\}$ such that every $m$ has $<k$ solutions to $m=a_1a_2$ with $a_1<a_2\in A$. Is it true that\[g_3(n)=\frac{\log\log n}{\log n}n+(c+o(1))\frac{n}{\log n}\]for some constant $c$?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/796,
erdos-80,open,needs-theory,mathematics / graph theory,"Let $c>0$ and let $f_c(n)$ be the maximal $m$ such that every graph $G$ with $n$ vertices and at least $cn^2$ edges, where each edge is contained in at least one triangle, must contain a book of size $m$, that is, an edge shared by at least $m$ different triangles. Estimate $f_c(n)$. In particular, is it true that $f_c(n)>n^{\epsilon}$ for some $\epsilon>0$? Or $f_c(n)\gg \log n$?","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/80,
erdos-802,open,needs-theory,mathematics / graph theory,Is it true that any $K_r$-free graph on $n$ vertices with average degree $t$ contains an independent set on\[\gg_r \frac{\log t}{t}n\]many vertices?,Erdős problems database (status: open; prize: no; tags: graph theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/802,
erdos-805,open,needs-theory,mathematics / graph theory,"For which functions $g(n)$ with $n>g(n)\geq (\log n)^2$ is there a graph on $n$ vertices in which every induced subgraph on $g(n)$ vertices contains a clique of size $\geq \log n$ and an independent set of size $\geq \log n$? In particular, is there such a graph for $g(n)=(\log n)^3$?",Erdős problems database (status: open; prize: no; tags: graph theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/805,
erdos-809,open,needs-theory,mathematics / graph theory,"Define the anti-Ramsey number $\chi_S(n,e,G)$ as the smallest $r$ such that there is a graph with $n$ vertices and $e$ edges with an $r$-colouring of its edges in which every copy of $G$ has entirely distinct edge colours. Is it true that, for all $k\geq 3$,\[\chi_S(n, \lfloor n^2/4\rfloor+1,C_{2k+1})\sim n^2/8?\]","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/809,
erdos-81,open,needs-theory,mathematics / graph theory,"Let $G$ be a chordal graph on $n$ vertices - that is, $G$ has no induced cycles of length greater than $3$. Can the edges of $G$ be partitioned into $n^2/6+O(n)$ many cliques?",Erdős problems database (status: open; prize: no; tags: graph theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/81,
erdos-810,open,needs-theory,mathematics / graph theory,"Does there exist some $\epsilon>0$ such that, for all sufficiently large $n$, there exists a graph $G$ on $n$ vertices with at least $\epsilon n^2$ many edges such that the edges can be coloured with $n$ colours so that every $C_4$ receives $4$ distinct colours?","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/810,
erdos-811,open,needs-theory,mathematics / graph theory,"Suppose $n\equiv 1\pmod{m}$. We say that an edge-colouring of $K_n$ using $m$ colours is balanced if every vertex sees exactly $\lfloor n/m\rfloor$ many edges of each colours. For which graphs $G$ is it true that, if $m=e(G)$, for all large $n\equiv 1\pmod{m}$, every balanced edge-colouring of $K_n$ with $m$ colours contains a rainbow copy of $G$? (That is, a subgraph isomorphic to $G$ where each edge receives a different colour.)","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/811,
erdos-812,open,needs-theory,mathematics / graph theory,"Is it true that\[\frac{R(n+1)}{R(n)}\geq 1+c\]for some constant $c>0$, for all large $n$? Is it true that\[R(n+1)-R(n) \gg n^2?\]","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A059442 and check the conjectured growth against them.,https://www.erdosproblems.com/812,
erdos-813,open,needs-theory,mathematics / graph theory,"Let $h(n)$ be minimal such that every graph on $n$ vertices where every set of $7$ vertices contains a triangle (a copy of $K_3$) must contain a clique on at least $h(n)$ vertices. Estimate $h(n)$ - in particular, do there exist constants $c_1,c_2>0$ such that\[n^{1/3+c_1}\ll h(n) \ll n^{1/2-c_2}?\]",Erdős problems database (status: open; prize: no; tags: graph theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/813,
erdos-817,open,needs-theory,mathematics / additive combinatorics,"Let $k\geq 3$ and define $g_k(n)$ to be the minimal $N$ such that $\{1,\ldots,N\}$ contains some $A$ of size $\lvert A\rvert=n$ such that\[\langle A\rangle = \left\{\sum_{a\in A}\epsilon_aa: \epsilon_a\in \{0,1\}\right\}\]contains no non-trivial $k$-term arithmetic progression. Estimate $g_k(n)$. In particular, is it true that\[g_3(n) \gg 3^n?\]",Erdős problems database (status: open; prize: no; tags: additive combinatorics),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/817,
erdos-819,open,needs-theory,mathematics / additive combinatorics,"Let $f(N)$ be maximal such that there exists $A\subseteq \{1,\ldots,N\}$ with $\lvert A\rvert=\lfloor N^{1/2}\rfloor$ such that $\lvert (A+A)\cap [1,N]\rvert=f(N)$. Estimate $f(N)$.",Erdős problems database (status: open; prize: no; tags: additive combinatorics),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/819,
erdos-82,open,needs-theory,mathematics / graph theory,Let $F(n)$ be maximal such that every graph on $n$ vertices contains a regular induced subgraph on at least $F(n)$ vertices. Prove that $F(n)/\log n\to \infty$.,Erdős problems database (status: open; prize: no; tags: graph theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A120414, A390256, A390257, A390919, A392636, A394400, A394462, A394539, A394563, A394564, A394573, A394574, A394930, A394933 and check the conjectured growth against them.",https://www.erdosproblems.com/82,
erdos-820,open,needs-theory,mathematics / number theory,"Let $H(n)$ be the smallest integer $l$ such that there exist $k<l$ with $(k^n-1,l^n-1)=1$. Is it true that $H(n)=3$ infinitely often? (That is, $(2^n-1,3^n-1)=1$ infinitely often?) Estimate $H(n)$. Is it true that there exists some constant $c>0$ such that, for all $\epsilon>0$,\[H(n) > \exp(n^{(c-\epsilon)/\log\log n})\]for infinitely many $n$ and\[H(n) < \exp(n^{(c+\epsilon)/\log\log n})\]for all large enough $n$? Does a similar upper bound hold for the smallest $k$ such that $(k^n-1,2^n-1)=1$?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A263647 and check the conjectured growth against them.,https://www.erdosproblems.com/820,
erdos-821,open,needs-theory,mathematics / number theory,"Let $g(n)$ count the number of $m$ such that $\phi(m)=n$. Is it true that, for every $\epsilon>0$, there exist infinitely many $n$ such that\[g(n) > n^{1-\epsilon}?\]",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A014197 and check the conjectured growth against them.,https://www.erdosproblems.com/821,
erdos-824,open,needs-theory,mathematics / number theory,"Let $h(x)$ count the number of integers $1\leq a<b<x$ such that $(a,b)=1$ and $\sigma(a)=\sigma(b)$, where $\sigma$ is the sum of divisors function. Is it true that $h(x)>x^{2-o(1)}$?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/824,
erdos-826,open,needs-theory,mathematics / number theory,"Are there infinitely many $n$ such that, for all $k\geq 1$,\[\tau(n+k)\ll k?\]",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/826,
erdos-827,open,needs-theory,mathematics / geometry,Let $n_k$ be minimal such that if $n_k$ points in $\mathbb{R}^2$ are in general position then there exists a subset of $k$ points such that all $\binom{k}{3}$ triples determine circles of different radii. Determine $n_k$.,Erdős problems database (status: open; prize: no; tags: geometry),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/827,
erdos-828,open,needs-theory,mathematics / number theory,"Is it true that, for any $a\in\mathbb{Z}$, there are infinitely many $n$ such that\[\phi(n) \mid n+a?\]",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/828,
erdos-829,open,needs-theory,mathematics / number theory,Let $A\subset\mathbb{N}$ be the set of cubes. Is it true that\[1_A\ast 1_A(n) \ll (\log n)^{O(1)}?\],Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/829,
erdos-830,open,needs-theory,mathematics / number theory,"We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$. Are there infinitely many amicable pairs? If $A(x)$ counts the number of amicable $1\leq a\leq b\leq x$ then is it true that\[A(x)>x^{1-o(1)}?\]",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A259180 and check the conjectured growth against them.,https://www.erdosproblems.com/830,
erdos-831,open,needs-theory,mathematics / geometry,Let $h(n)$ be maximal such that in any $n$ points in $\mathbb{R}^2$ (with no three on a line and no four on a circle) there are at least $h(n)$ many circles of different radii passing through three points. Estimate $h(n)$.,Erdős problems database (status: open; prize: no; tags: geometry),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/831,
erdos-835,open,compute-checkable-small-cases,mathematics / graph theory,"Open, but could be proved with a finite example. Does there exist a $k>2$ such that the $k$-sized subsets of $\{1,\ldots,2k\}$ can be coloured with $k+1$ colours such that for every $A\subset \{1,\ldots,2k\}$ with $\lvert A\rvert=k+1$ all $k+1$ colours appear among the $k$-sized subsets of $A$?","Erdős problems database (status: verifiable; prize: no; tags: graph theory, hypergraphs)",The database marks this as provable by a finite computation if true: run the verification. Extend OEIS N/A first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/835,
erdos-836,open,needs-theory,mathematics / graph theory,"Let $r\geq 2$ and $G$ be a $r$-uniform hypergraph with chromatic number $3$ (that is, there is a $3$-colouring of the vertices of $G$ such that no edge is monochromatic). Suppose any two edges of $G$ have a non-empty intersection. Must $G$ contain $O(r^2)$ many vertices? Must there be two edges which meet in $\gg r$ many vertices?","Erdős problems database (status: open; prize: no; tags: graph theory, hypergraphs, chromatic number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/836,
erdos-837,open,needs-theory,mathematics / graph theory,"Let $k\geq 2$ and $A_k\subseteq [0,1]$ be the set of $\alpha$ such that there exists some $\beta(\alpha)>\alpha$ with the property that, if $G_1,G_2,\ldots$ is a sequence of $k$-uniform hypergraphs with\[\liminf \frac{e(G_n)}{\binom{\lvert G_n\rvert}{k}} >\alpha\]then there exist subgraphs $H_n\subseteq G_n$ such that $\lvert H_n\rvert \to \infty$ and\[\liminf \frac{e(H_n)}{\binom{\lvert H_n\rvert}{k}} >\beta,\]and further that this property does not necessarily hold if $>\alpha$ is replaced by $\geq \alpha$. What is $A_3$?","Erdős problems database (status: open; prize: no; tags: graph theory, hypergraphs)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/837,
erdos-838,open,needs-theory,mathematics / geometry,"Let $f(n)$ be maximal such that any $n$ points in $\mathbb{R}^2$, with no three on a line, determine at least $f(n)$ different convex subsets. Estimate $f(n)$ - in particular, does there exist a constant $c$ such that\[\lim \frac{\log f(n)}{(\log n)^2}=c?\]","Erdős problems database (status: open; prize: no; tags: geometry, convex)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/838,
erdos-839,open,needs-theory,mathematics / number theory,Let $1\leq a_1<a_2<\cdots$ be a sequence of integers such that no $a_i$ is the sum of consecutive $a_j$ for $j<i$. Is it true that\[\limsup \frac{a_n}{n}=\infty?\]Or even\[\lim \frac{1}{\log x}\sum_{a_n<x}\frac{1}{a_n}=0?\],Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/839,
erdos-84,open,needs-theory,mathematics / graph theory,"The cycle set of a graph $G$ on $n$ vertices is a set $A\subseteq \{3,\ldots,n\}$ such that there is a cycle in $G$ of length $\ell$ if and only if $\ell \in A$. Let $f(n)$ count the number of possible such $A$. Prove that $f(n)=o(2^n)$. Prove that $f(n)/2^{n/2}\to \infty$.","Erdős problems database (status: open; prize: no; tags: graph theory, cycles)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/84,
erdos-840,open,needs-theory,mathematics / additive combinatorics,"Let $f(N)$ be the size of the largest quasi-Sidon subset $A\subset\{1,\ldots,N\}$, where we say that $A$ is quasi-Sidon if\[\lvert A+A\rvert=(1+o(1))\binom{\lvert A\rvert}{2}.\]How does $f(N)$ grow?","Erdős problems database (status: open; prize: no; tags: additive combinatorics, sidon sets)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/840,
erdos-848,open,compute-checkable-small-cases,mathematics / number theory,"Resolved up to a finite check. Is the maximum size of a set $A\subseteq \{1,\ldots,N\}$ such that $ab+1$ is never squarefree (for all $a,b\in A$) achieved by taking those $n\equiv 7\pmod{25}$?",Erdős problems database (status: decidable; prize: no; tags: number theory),The database marks this as reduced to a finite computation: run it. Extend OEIS N/A first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/848,
erdos-849,open,needs-theory,mathematics / number theory,"Is it true that, for every integer $t\geq 1$, there is some integer $a$ such that\[\binom{n}{k}=a\](with $1\leq k\leq n/2$) has exactly $t$ solutions?","Erdős problems database (status: open; prize: no; tags: number theory, binomial coefficients)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A003016, A003015, A059233, A098565, A090162, A180058, A182237 and check the conjectured growth against them.",https://www.erdosproblems.com/849,
erdos-85,open,needs-theory,mathematics / graph theory,"Let $n\geq 4$ and $f(n)$ be minimal such that every graph on $n$ vertices with minimal degree $\geq f(n)$ contains a $C_4$. Is it true that, for all large $n$, $f(n+1)\geq f(n)$?",Erdős problems database (status: open; prize: no; tags: graph theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A006672, possible and check the conjectured growth against them.",https://www.erdosproblems.com/85,
erdos-850,open,needs-theory,mathematics / number theory,"Can there exist two distinct integers $x$ and $y$ such that $x,y$ have the same prime factors, $x+1,y+1$ have the same prime factors, and $x+2,y+2$ also have the same prime factors?","Erdős problems database (status: open; prize: no; tags: number theory, primes)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A343101 and check the conjectured growth against them.,https://www.erdosproblems.com/850,
erdos-852,open,needs-theory,mathematics / number theory,"Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Let $h(x)$ be maximal such that for some $n<x$ the numbers $d_n,d_{n+1},\ldots,d_{n+h(x)-1}$ are all distinct. Estimate $h(x)$. In particular, is it true that\[h(x) >(\log x)^c\]for some constant $c>0$, and\[h(x)=o(\log x)?\]","Erdős problems database (status: open; prize: no; tags: number theory, primes)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A001223, A053597, A078515 and check the conjectured growth against them.",https://www.erdosproblems.com/852,
erdos-853,open,needs-theory,mathematics / number theory,"Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Let $r(x)$ be the smallest even integer $t$ such that $d_n=t$ has no solutions for $n\leq x$. Is it true that $r(x)\to \infty$? Or even $r(x)/\log x \to \infty$?","Erdős problems database (status: open; prize: no; tags: number theory, primes)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A001223, A390769 and check the conjectured growth against them.",https://www.erdosproblems.com/853,
erdos-854,open,needs-theory,mathematics / number theory,"Let $n_k$ denote the $k$th primorial, i.e. the product of the first $k$ primes. If $1=a_1<a_2<\cdots a_{\phi(n_k)}=n_k-1$ is the sequence of integers coprime to $n_k$, then estimate the smallest even integer not of the form $a_{i+1}-a_i$. Are there\[\gg \max_i (a_{i+1}-a_i)\]many even integers of the form $a_{j+1}-a_j$?",Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A389839, A048670 and check the conjectured growth against them.",https://www.erdosproblems.com/854,
erdos-855,open,needs-theory,mathematics / number theory,"If $\pi(x)$ counts the number of primes in $[1,x]$ then is it true that (for large $x$ and $y$)\[\pi(x+y) \leq \pi(x)+\pi(y)?\]","Erdős problems database (status: open; prize: no; tags: number theory, primes)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A023193 and check the conjectured growth against them.,https://www.erdosproblems.com/855,
erdos-856,open,needs-theory,mathematics / number theory,"Let $k\geq 3$ and $f_k(N)$ be the maximum value of $\sum_{n\in A}\frac{1}{n}$, where $A$ ranges over all subsets of $\{1,\ldots,N\}$ which contain no subset of size $k$ with the same pairwise least common multiple. Estimate $f_k(N)$.",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/856,
erdos-857,open,needs-theory,mathematics / combinatorics,"Let $m=m(n,k)$ be minimal such that in any collection of sets $A_1,\ldots,A_m\subseteq \{1,\ldots,n\}$ there must exist a sunflower of size $k$ - that is, some collection of $k$ of the $A_i$ which pairwise have the same intersection. Estimate $m(n,k)$, or even better, give an asymptotic formula.",Erdős problems database (status: open; prize: no; tags: combinatorics),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/857,
erdos-859,open,needs-theory,mathematics / number theory,"Let $t\geq 1$ and let $d_t$ be the density of the set of integers $n\in\mathbb{N}$ for which $t$ can be represented as the sum of distinct divisors of $n$. Do there exist constants $c_1,c_2>0$ such that\[d_t \sim \frac{c_1}{(\log t)^{c_2}}\]as $t\to \infty$?","Erdős problems database (status: open; prize: no; tags: number theory, divisors)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/859,
erdos-86,open,needs-theory,mathematics / graph theory,Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Is it true that every subgraph of $Q_n$ with\[\geq \left(\frac{1}{2}+o(1)\right)n2^{n-1}\]many edges contains a $C_4$?,Erdős problems database (status: open; prize: $100; tags: graph theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A245762 and check the conjectured growth against them.,https://www.erdosproblems.com/86,
erdos-860,open,needs-theory,mathematics / number theory,"Let $h(n)$ be such that, for any $m\geq 1$, in the interval $(m,m+h(n))$ there exist distinct integers $a_i$ for $1\leq i\leq \pi(n)$ such that $p_i\mid a_i$, where $p_i$ denotes the $i$th prime. Estimate $h(n)$.","Erdős problems database (status: open; prize: no; tags: number theory, primes)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A048670, A058989 and check the conjectured growth against them.",https://www.erdosproblems.com/860,
erdos-864,open,needs-theory,mathematics / number theory,"Let $A\subseteq \{1,\ldots N\}$ be a set such that there exists at most one $n$ with more than one solution to $n=a+b$ (with $a\leq b\in A$). Estimate the maximal possible size of $\lvert A\rvert$ - in particular, is it true that\[\lvert A\rvert \leq (1+o(1))\frac{2}{\sqrt{3}}N^{1/2}?\]","Erdős problems database (status: open; prize: no; tags: number theory, sidon sets, additive combinatorics)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A389182 and check the conjectured growth against them.,https://www.erdosproblems.com/864,
erdos-866,open,needs-theory,mathematics / number theory,"Let $k\geq 3$ and $g_k(N)$ be minimal such that if $A\subseteq \{1,\ldots,2N\}$ has $\lvert A\rvert \geq N+g_k(N)$ then there exist integers $b_1,\ldots,b_k$ such that all $\binom{k}{2}$ pairwise sums are in $A$ (but the $b_i$ themselves need not be in $A$). Estimate $g_k(N)$.","Erdős problems database (status: open; prize: no; tags: number theory, additive combinatorics)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/866,
erdos-87,open,needs-theory,mathematics / graph theory,"Let $\epsilon >0$. Is it true that, if $k$ is sufficiently large, then\[R(G)>(1-\epsilon)^kR(k)\]for every graph $G$ with chromatic number $\chi(G)=k$? Even stronger, is there some $c>0$ such that, for all large $k$, $R(G)>cR(k)$ for every graph $G$ with chromatic number $\chi(G)=k$?","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A059442, possible and check the conjectured growth against them.",https://www.erdosproblems.com/87,
erdos-870,open,needs-theory,mathematics / number theory,Let $k\geq 3$ and $A$ be an additive basis of order $k$. Does there exist a constant $c=c(k)>0$ such that if $r(n)\geq c\log n$ for all large $n$ then $A$ must contain a minimal basis of order $k$? (Here $r(n)$ counts the number of representations of $n$ as the sum of at most $k$ elements from $A$.),"Erdős problems database (status: open; prize: no; tags: number theory, additive basis)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/870,
erdos-872,open,needs-theory,mathematics / number theory,"Consider the two-player game in which players alternately choose integers from $\{2,3,\ldots,n\}$ to be included in some set $A$ (the same set for both players) such that no $a\mid b$ for $a\neq b\in A$. The game ends when no legal move is possible. One player wants the game to last as long as possible, the other wants the game to end quickly. How long can the game be guaranteed to last for? At least $\epsilon n$ moves? (For $\epsilon>0$ and $n$ sufficiently large.) At least $(1-\epsilon)\frac{n}{2}$ moves?","Erdős problems database (status: open; prize: no; tags: number theory, primitive sets)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/872,
erdos-873,open,needs-theory,mathematics / number theory,"Let $A=\{a_1<a_2<\cdots\}\subseteq \mathbb{N}$ and let $F(A,X,k)$ count the number of $i$ such that\[[a_i,a_{i+1},\ldots,a_{i+k-1}] < X,\]where the left-hand side is the least common multiple. Is it true that, for every $\epsilon >0$, there exists some $k$ such that\[F(A,X,k)<X^\epsilon?\]",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/873,
erdos-875,open,needs-theory,mathematics / additive combinatorics,"Let $A=\{a_1<a_2<\cdots\}\subset \mathbb{N}$ be an infinite set such that the sets\[S_r = \{ a_1+\cdots +a_r : a_1<\cdots<a_r\in A\}\]are disjoint for distinct $r\geq 1$. How fast can such a sequence grow? How small can $a_{n+1}-a_n$ be? In particular, for which $c$ is it possible that $a_{n+1}-a_n\leq n^{c}$?",Erdős problems database (status: open; prize: no; tags: additive combinatorics),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/875,
erdos-876,open,needs-theory,mathematics / additive combinatorics,"Let $A=\{a_1<a_2<\cdots\}\subset \mathbb{N}$ be an infinite sum-free set - that is, there are no solutions to\[a=b_1+\cdots+b_r\]with $b_1<\cdots<b_r<a\in A$. How small can $a_{n+1}-a_n$ be? Is it possible that $a_{n+1}-a_n<n$?",Erdős problems database (status: open; prize: no; tags: additive combinatorics),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/876,
erdos-878,open,needs-theory,mathematics / number theory,"If $n=\prod_{1\leq i\leq t} p_i^{k_i}$ is the factorisation of $n$ into distinct primes then let\[f(n)=\sum p_i^{\ell_i},\]where $\ell_i$ is chosen such that $n\in [p_i^{\ell_i},p_i^{\ell_i+1})$. Furthermore, let\[F(n)=\max \sum_{i} a_i\]where the maximum is taken over all distinct $a_1,\ldots,a_k\leq n$ such that $(a_i,a_j)=1$ for $i\neq j$ and all prime factors of each $a_i$ are prime factors of $n$. Is it true that, for almost all $n$,\[f(n)=o(n\log\log n)\]and\[F(n) \gg n\log\log n?\]Is it true that\[\max_{n\leq x}f(n)\sim \frac{x\log x}{\log\log x}?\]Is it true that (for all $x$, or perhaps just for all large $x$)\[\max_{n\leq x}f(n)=\max_{n\leq x}F(n)?\]Find an asymptotic formula for the number of $n<x$ such that $f(n)=F(n)$. Find an asymptotic formula for\[H(x)=\sum_{n<x}\frac{f(n)}{n}.\]Is it true that\[H(x) \ll x\log\log\log\log x?\]",Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A339378, possible and check the conjectured growth against them.",https://www.erdosproblems.com/878,
erdos-879,open,needs-theory,mathematics / number theory,"Call a set $S\subseteq \{1,\ldots,n\}$ admissible if $(a,b)=1$ for all $a\neq b\in S$. Let\[G(n) = \max_{S\subseteq \{1,\ldots,n\}} \sum_{a\in S}a\]and\[H(n)=\sum_{p<n}p+ n\pi(n^{1/2}).\]Is it true that\[G(n) >H(n)-n^{1+o(1)}?\]Is it true that, for every $k\geq 2$, if $n$ is sufficiently large then the admissible set which maximises $G(n)$ contains at least one integer with at least $k$ prime factors?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A186736 and check the conjectured growth against them.,https://www.erdosproblems.com/879,
erdos-881,open,needs-theory,mathematics / number theory,"Let $A\subset\mathbb{N}$ be an additive basis of order $k$ which is minimal, in the sense that if $B\subset A$ is any infinite set then $A\backslash B$ is not a basis of order $k$. Must there exist an infinite $B\subset A$ such that $A\backslash B$ is a basis of order $k+1$?","Erdős problems database (status: open; prize: no; tags: number theory, additive basis)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/881,
erdos-883,open,needs-theory,mathematics / number theory,"For $A\subseteq \{1,\ldots,n\}$ let $G(A)$ be the graph with vertex set $A$, where two integers are joined by an edge if they are coprime. Is it true that if\[\lvert A\rvert >\lfloor\tfrac{n}{2}\rfloor+\lfloor\tfrac{n}{3}\rfloor-\lfloor\tfrac{n}{6}\rfloor\]then $G(A)$ contains all odd cycles of length $\leq \frac{n}{3}+1$? Is it true that, for every $\ell\geq 1$, if $n$ is sufficiently large and\[\lvert A\rvert >\lfloor\tfrac{n}{2}\rfloor+\lfloor\tfrac{n}{3}\rfloor-\lfloor\tfrac{n}{6}\rfloor\]then $G(A)$ must contain a complete $(1,\ell,\ell)$ triparite graph on $2\ell+1$ vertices?","Erdős problems database (status: open; prize: no; tags: number theory, graph theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/883,
erdos-885,open,needs-theory,mathematics / number theory,"For integer $n\geq 1$ we define the factor difference set of $n$ by\[D(n) = \{\lvert a-b\rvert : n=ab\}.\]Is it true that, for every $k\geq 1$, there exist integers $N_1<\cdots<N_k$ such that\[\lvert \cap_i D(N_i)\rvert \geq k?\]","Erdős problems database (status: open; prize: no; tags: number theory, divisors)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/885,
erdos-886,open,needs-theory,mathematics / number theory,"Let $\epsilon>0$. Is it true that, for all large $n$, the number of divisors of $n$ in $(n^{1/2},n^{1/2}+n^{1/2-\epsilon})$ is $O_\epsilon(1)$?","Erdős problems database (status: open; prize: no; tags: number theory, divisors)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/886,
erdos-887,open,needs-theory,mathematics / number theory,"Is there an absolute constant $K$ such that, for every $C>0$, if $n$ is sufficiently large then $n$ has at most $K$ divisors in $(n^{1/2},n^{1/2}+C n^{1/4})$.","Erdős problems database (status: open; prize: no; tags: number theory, divisors)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/887,
erdos-889,open,needs-theory,mathematics / number theory,"For $k\geq 0$ and $n\geq 1$ let $v(n,k)$ count the prime factors of $n+k$ which do not divide $n+i$ for $0\leq i<k$. Equivalently, $v(n,k)$ counts the number of prime factors of $n+k$ which are $>k$. Is it true that\[v_0(n)=\max_{k\geq 0}v(n,k)\to \infty\]as $n\to \infty$?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/889,
erdos-89,open,needs-theory,mathematics / geometry,Does every set of $n$ distinct points in $\mathbb{R}^2$ determine $\gg n/\sqrt{\log n}$ many distinct distances?,"Erdős problems database (status: open; prize: $500; tags: geometry, distances)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A186704, A131628 and check the conjectured growth against them.",https://www.erdosproblems.com/89,
erdos-890,open,needs-theory,mathematics / number theory,"If $\omega_k(n)$ counts the number of distinct prime factors of $n$ which are $>k$, then is it true that, for every $k\geq 1$,\[\liminf_{n\to \infty}\sum_{0\leq i<k}\omega_k(n+i)\leq k?\]Is it true that\[\limsup_{n\to \infty}\left(\sum_{0\leq i<k}\omega(n+i)\right) \frac{\log\log n}{\log n}=1,\]where $\omega$ counts the number of distinct prime factors without restriction?","Erdős problems database (status: open; prize: no; tags: number theory, primes)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/890,
erdos-891,open,needs-theory,mathematics / number theory,"Let $2=p_1<p_2<\cdots$ be the primes and $k\geq 2$. Is it true that, for all sufficiently large $n$, there must exist an integer in $[n,n+p_1\cdots p_k)$ with $>k$ many prime factors?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/891,
erdos-892,open,needs-theory,mathematics / number theory,"Is there a necessary and sufficient condition for a sequence of integers $b_1<b_2<\cdots$ that ensures there exists a primitive sequence $a_1<a_2<\cdots$ (i.e. no element divides another) with $a_n \ll b_n$ for all $n$? In particular, is this always possible if there are no non-trivial solutions to $(b_i,b_j)=b_k$? Similarly, find necessary and sufficient conditions on a sequence $n_1<n_2<\cdots$ that ensure there exists a primitive set $A$ such that\[\lvert A\cap [1,2^{n_i}]\rvert \gg 2^{n_i}\]for every $i$.","Erdős problems database (status: open; prize: no; tags: number theory, primitive sets)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/892,
erdos-893,open,needs-theory,mathematics / number theory,If $\tau(n)$ counts the divisors of $n$ then let\[f(n)=\sum_{1\leq k\leq n}\tau(2^k-1).\]Does $f(2n)/f(n)$ tend to a limit?,"Erdős problems database (status: open; prize: no; tags: number theory, divisors)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A046801, possible and check the conjectured growth against them.",https://www.erdosproblems.com/893,
erdos-9,open,needs-theory,mathematics / number theory,"Let $A$ be the set of all odd integers $\geq 1$ not of the form $p+2^{k}+2^l$ (where $k,l\geq 0$ and $p$ is prime). Is the upper density of $A$ positive?","Erdős problems database (status: open; prize: no; tags: number theory, additive basis, primes)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A006286 and check the conjectured growth against them.,https://www.erdosproblems.com/9,
erdos-901,open,needs-theory,mathematics / combinatorics,Let $m(n)$ be minimal such that there is an $n$-uniform hypergraph with $m(n)$ edges which is $3$-chromatic. Estimate $m(n)$.,"Erdős problems database (status: open; prize: no; tags: combinatorics, hypergraphs)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/901,
erdos-902,open,needs-theory,mathematics / graph theory,Let $f(n)$ be minimal such that there is a tournament (a complete directed graph) on $f(n)$ vertices such that every set of $n$ vertices is dominated by at least one other vertex. Estimate $f(n)$.,Erdős problems database (status: open; prize: no; tags: graph theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A362137 and check the conjectured growth against them.,https://www.erdosproblems.com/902,
erdos-906,open,needs-theory,mathematics / analysis,"Is there an entire non-zero function $f:\mathbb{C}\to \mathbb{C}$ such that, for any infinite sequence $n_1<n_2<\cdots$, the set\[\{ z: f^{(n_k)}(z)=0 \textrm{ for some }k\geq 1\}\]is everywhere dense?","Erdős problems database (status: open; prize: no; tags: analysis, iterated functions)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/906,
erdos-91,open,needs-theory,mathematics / geometry,Let $n$ be a sufficiently large integer. Suppose $A\subset \mathbb{R}^2$ has $\lvert A\rvert=n$ and minimises the number of distinct distances between points in $A$. Prove that there are at least two (and probably many) such $A$ which are non-similar.,"Erdős problems database (status: open; prize: no; tags: geometry, distances)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A186704, possible and check the conjectured growth against them.",https://www.erdosproblems.com/91,
erdos-911,open,needs-theory,mathematics / graph theory,"Let $\hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges that is Ramsey for $G$. Is there a function $f$ such that $f(x)/x\to \infty$ as $x\to \infty$ such that, for all large $C$, if $G$ is a graph with $n$ vertices and $e\geq Cn$ edges then\[\hat{R}(G) > f(C) e?\]","Erdős problems database (status: open; prize: no; tags: graph theory, ramsey theory)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/911,
erdos-912,open,needs-theory,mathematics / number theory,If\[n! = \prod_i p_i^{k_i}\]is the factorisation into distinct primes then let $h(n)$ count the number of distinct exponents $k_i$. Prove that there exists some $c>0$ such that\[h(n) \sim c \left(\frac{n}{\log n}\right)^{1/2}\]as $n\to \infty$.,"Erdős problems database (status: open; prize: no; tags: number theory, factorials)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A071626 and check the conjectured growth against them.,https://www.erdosproblems.com/912,
erdos-913,open,needs-theory,mathematics / number theory,Are there infinitely many $n$ such that if\[n(n+1) = \prod_i p_i^{k_i}\]is the factorisation into distinct primes then all exponents $k_i$ are distinct?,Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A359747 and check the conjectured growth against them.,https://www.erdosproblems.com/913,
erdos-917,open,needs-theory,mathematics / graph theory,"Let $k\geq 4$ and $f_k(n)$ be the largest number of edges in a graph on $n$ vertices which has chromatic number $k$ and is critical (i.e. deleting any edge reduces the chromatic number). Is it true that\[f_k(n) \gg_k n^2?\]Is it true that\[f_6(n)\sim n^2/4?\]More generally, is it true that, for $k\geq 6$,\[f_k(n) \sim \frac{1}{2}\left(1-\frac{1}{\lfloor k/3\rfloor}\right)n^2?\]","Erdős problems database (status: open; prize: no; tags: graph theory, chromatic number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/917,
erdos-918,open,needs-theory,mathematics / graph theory,Is there a graph with $\aleph_2$ vertices and chromatic number $\aleph_2$ such that every subgraph on $\aleph_1$ vertices has chromatic number $\leq\aleph_0$? Is there a graph with $\aleph_{\omega+1}$ vertices and chromatic number $\aleph_1$ such that every subgraph on $\aleph_\omega$ vertices has chromatic number $\leq\aleph_0$?,"Erdős problems database (status: open; prize: no; tags: graph theory, chromatic number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/918,
erdos-919,open,needs-theory,mathematics / graph theory,Is there a graph $G$ with vertex set $\omega_2^2$ and chromatic number $\aleph_2$ such that every subgraph whose vertices have a lesser type has chromatic number $\leq \aleph_0$? What if instead we ask for $G$ to have chromatic number $\aleph_1$?,"Erdős problems database (status: open; prize: no; tags: graph theory, chromatic number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/919,
erdos-928,open,needs-theory,mathematics / number theory,"Let $\alpha,\beta\in (0,1)$ and let $P(n)$ denote the largest prime divisor of $n$. Does the density of integers $n$ such that $P(n)<n^{\alpha}$ and $P(n+1)<(n+1)^\beta$ exist?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A006530 and check the conjectured growth against them.,https://www.erdosproblems.com/928,
erdos-929,open,needs-theory,mathematics / number theory,"Let $k\geq 2$ be large and let $S(k)$ be the minimal $x$ such that there is a positive density set of $n$ where\[n+1,n+2,\ldots,n+k\]are all divisible by primes $\leq x$. Estimate $S(k)$ - in particular, is it true that $S(k)\geq k^{1-o(1)}$?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/929,
erdos-930,open,needs-theory,mathematics / number theory,"Is it true that, for every $r$, there is a $k$ such that if $I_1,\ldots,I_r$ are disjoint intervals of consecutive integers, all of length at least $k$, then\[\prod_{1\leq i\leq r}\prod_{m\in I_i}m\]is not a perfect power?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/930,
erdos-931,open,needs-theory,mathematics / number theory,Let $k_1\geq k_2\geq 3$. Are there only finitely many $n_2\geq n_1+k_1$ such that\[\prod_{1\leq i\leq k_1}(n_1+i)\textrm{ and }\prod_{1\leq j\leq k_2}(n_2+j)\]have the same prime factors?,Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/931,
erdos-932,open,needs-theory,mathematics / number theory,Let $p_k$ denote the $k$th prime. For infinitely many $r$ there are at least two integers $p_r<n<p_{r+1}$ all of whose prime factors are $<p_{r+1}-p_r$.,Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A387864 and check the conjectured growth against them.,https://www.erdosproblems.com/932,
erdos-933,open,needs-theory,mathematics / number theory,"If $n(n+1)=2^k3^lm$, where $(m,6)=1$, then is it true that\[\limsup_{n\to \infty} \frac{2^k3^l}{n\log n}=\infty?\]",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/933,
erdos-934,open,needs-theory,mathematics / graph theory,Let $h_t(d)$ be minimal such that every graph $G$ with $h_t(d)$ edges and maximal degree $\leq d$ contains two edges whose shortest path between them has length $\geq t$. Estimate $h_t(d)$.,Erdős problems database (status: open; prize: no; tags: graph theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/934,
erdos-935,open,needs-theory,mathematics / number theory,"For any integer $n=\prod p^{k_p}$ let $Q_2(n)$ be the powerful part of $n$, so that\[Q_2(n) = \prod_{\substack{p\\ k_p\geq 2}}p^{k_p}.\]Is it true that, for every $\epsilon>0$ and $\ell\geq 1$, if $n$ is sufficiently large then\[Q_2(n(n+1)\cdots(n+\ell))<n^{2+\epsilon}?\]If $\ell\geq 2$ then is\[\limsup_{n\to \infty}\frac{Q_2(n(n+1)\cdots(n+\ell))}{n^2}\]infinite? If $\ell\geq 2$ then is\[\lim_{n\to \infty}\frac{Q_2(n(n+1)\cdots(n+\ell))}{n^{\ell+1}}=0?\]",Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A057521, A389244, possible and check the conjectured growth against them.",https://www.erdosproblems.com/935,
erdos-936,open,needs-theory,mathematics / number theory,Are\[2^n\pm 1\]and\[n!\pm 1\]powerful (i.e. if $p\mid m$ then $p^2\mid m$) for only finitely many $n$?,Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A146968, possible and check the conjectured growth against them.",https://www.erdosproblems.com/936,
erdos-938,open,needs-theory,mathematics / number theory,"Let $A=\{n_1<n_2<\cdots\}$ be the sequence of powerful numbers (if $p\mid n$ then $p^2\mid n$). Are there only finitely many three-term progressions of consecutive terms $n_k,n_{k+1},n_{k+2}$?",Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A001694, A076446, possible and check the conjectured growth against them.",https://www.erdosproblems.com/938,
erdos-939,open,needs-theory,mathematics / number theory,"Let $r\geq 2$. An $r$-powerful number $n$ is one such that if $p\mid n$ then $p^r\mid n$. If $r\geq 4$ then can the sum of $r-2$ coprime $r$-powerful numbers ever be itself $r$-powerful? Are there at most finitely many such solutions? Are there infinitely many triples of coprime $3$-powerful numbers $a,b,c$ such that $a+b=c$?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/939,
erdos-940,open,needs-theory,mathematics / number theory,Let $r\geq 3$. A number $n$ is $r$-powerful if for every prime $p$ which divides $n$ we have $p^r\mid n$. Are there infinitely many integers which are not the sum of at most $r$ many $r$-powerful numbers? Does the set of integers which are the sum of at most $r$ $r$-powerful numbers have density $0$?,Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/940,
erdos-942,open,needs-theory,mathematics / number theory,"Let $h(n)$ count the number of powerful (if $p\mid m$ then $p^2\mid m$) integers in $[n^2,(n+1)^2)$. Estimate $h(n)$. In particular is there some constant $c>0$ such that\[h(n) < (\log n)^{c+o(1)}\]and, for infinitely many $n$,\[h(n) >(\log n)^{c-o(1)}?\]",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/942,
erdos-943,open,needs-theory,mathematics / number theory,Let $A$ be the set of powerful numbers (if $p\mid n$ then $p^2\mid n$). Is it true that\[1_A\ast 1_A(n)=n^{o(1)}\]for every $n$?,Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/943,
erdos-944,open,needs-theory,mathematics / graph theory,"A critical vertex, edge, or set of edges, is one whose deletion lowers the chromatic number. Let $k\geq 4$ and $r\geq 1$. Must there exist a graph $G$ with chromatic number $k$ such that every vertex is critical, yet every critical set of edges has size $>r$?","Erdős problems database (status: open; prize: no; tags: graph theory, chromatic number)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/944,
erdos-945,open,needs-theory,mathematics / number theory,"Let $F(x)$ be the maximal $k$ such that there exist $n+1,\ldots,n+k\leq x$ with $\tau(n+1),\ldots,\tau(n+k)$ all distinct (where $\tau(m)$ counts the divisors of $m$). Estimate $F(x)$. In particular, is it true that\[F(x) \leq (\log x)^{O(1)}?\]In other words, is there a constant $C>0$ such that, for all large $x$, every interval $[x,x+(\log x)^C]$ contains two integers with the same number of divisors?","Erdős problems database (status: open; prize: no; tags: number theory, divisors)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible, A048892 and check the conjectured growth against them.",https://www.erdosproblems.com/945,
erdos-949,open,needs-theory,mathematics / ramsey theory,Let $S\subset \mathbb{R}$ be a set containing no solutions to $a+b=c$. Must there be a set $A\subseteq \mathbb{R}\backslash S$ of cardinality continuum such that $A+A\subseteq \mathbb{R}\backslash S$?,Erdős problems database (status: open; prize: no; tags: ramsey theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/949,
erdos-950,open,needs-theory,mathematics / number theory,Let\[f(n) = \sum_{p<n}\frac{1}{n-p}.\]Is it true that\[\liminf f(n)=1\]and\[\limsup f(n)=\infty?\]Is it true that $f(n)=o(\log\log n)$ for all $n$?,"Erdős problems database (status: open; prize: no; tags: number theory, primes)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/950,
erdos-951,open,needs-theory,mathematics / number theory,"Let $1<a_1<\cdots$ be a sequence of real numbers such that\[\left\lvert \prod_i a_i^{k_i}-\prod_j a_j^{\ell_j}\right\rvert \geq 1\]for every distinct pair of non-negative finitely supported integer tuples $k_i,\ell_j\geq 0$. Is it true that\[\#\{ a_i \leq x\} \leq \pi(x)?\]",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/951,
erdos-952,open,needs-theory,mathematics / number theory,"Is there an infinite sequence of distinct Gaussian primes $x_1,x_2,\ldots$ such that\[\lvert x_{n+1}-x_n\rvert \ll 1?\]",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/952,
erdos-953,open,needs-theory,mathematics / geometry,"Let $A\subset \{ x\in \mathbb{R}^2 : \lvert x\rvert <r\}$ be a measurable set with no integer distances, that is, such that $\lvert a-b\rvert \not\in \mathbb{Z}$ for any distinct $a,b\in A$. How large can the measure of $A$ be?","Erdős problems database (status: open; prize: no; tags: geometry, distances)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/953,
erdos-954,open,needs-theory,mathematics / number theory,"Let $0=a_0<a_1<a_2<\cdots$ be the sequence of integers defined by $a_0=0$ and $a_1=1$, and $a_{k+1}$ is the smallest integer $n$ for which the number of solutions to $a_i+a_j \leq n$ (with $0\leq i\leq j\leq k$ and $j\geq 1$) is $<n$. Is the number of solutions to $a_i+a_j \leq x$ equal to $x+O(x^{1/4+o(1)})$?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A390642 and check the conjectured growth against them.,https://www.erdosproblems.com/954,
erdos-955,open,needs-theory,mathematics / number theory,Let\[s(n)=\sigma(n)-n=\sum_{\substack{d\mid n\\ d<n}}d\]be the sum of proper divisors function. If $A\subset \mathbb{N}$ has density $0$ then $s^{-1}(A)$ must also have density $0$.,Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/955,
erdos-956,open,needs-theory,mathematics / geometry,"If $C,D\subseteq \mathbb{R}^2$ then the distance between $C$ and $D$ is defined by\[\delta(C,D)=\inf_{\substack{c\in C\\ d\in D}}\| c-d\|.\]Let $h(n)$ be the maximal number of unit distances between disjoint convex translates. That is, the maximal $m$ such that there is a compact convex set $C\subset \mathbb{R}^2$ and a set $X$ of size $n$ such that all $(C+x)_{x\in X}$ are disjoint and there are $m$ pairs $x_1,x_2\in X$ such that\[\delta(C+x_1,C+x_2)=1.\]Determine $h(n)$ - in particular, prove that there exists a constant $c>0$ such that $h(n)>n^{1+c}$ for all large $n$.","Erdős problems database (status: open; prize: no; tags: geometry, distances, convex)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/956,
erdos-959,open,needs-theory,mathematics / geometry,"Let $A\subset \mathbb{R}^2$ be a set of size $n$ and let $\{d_1,\ldots,d_k\}$ be the set of distinct distances determined by $A$. Let $f(d)$ be the number of times the distance $d$ is determined, and suppose the $d_i$ are ordered such that\[f(d_1)\geq f(d_2)\geq \cdots \geq f(d_k).\]Estimate\[\max (f(d_1)-f(d_2)),\]where the maximum is taken over all $A$ of size $n$.","Erdős problems database (status: open; prize: no; tags: geometry, distances)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/959,
erdos-96,open,needs-theory,mathematics / geometry,If $n$ points in $\mathbb{R}^2$ form a convex polygon then there are $O(n)$ many pairs which are distance $1$ apart.,"Erdős problems database (status: open; prize: no; tags: geometry, distances, convex)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/96,
erdos-961,open,needs-theory,mathematics / number theory,Let $f(k)$ be the minimal $n$ such that every set of $n$ consecutive integers $>k$ contains an integer divisible by a prime $>k$. Estimate $f(k)$.,Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A213253 and check the conjectured growth against them.,https://www.erdosproblems.com/961,
erdos-962,open,needs-theory,mathematics / number theory,"Let $k(n)$ be the maximal $k$ such that there exists $m\leq n$ such that each of the integers\[m+1,\ldots,m+k\]are divisible by at least one prime $>k$. Estimate $k(n)$ - in particular, is it true that\[\log k(n) \leq (\log n)^{1/2+o(1)}?\]",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A327909 and check the conjectured growth against them.,https://www.erdosproblems.com/962,
erdos-963,open,needs-theory,mathematics / number theory,"Let $f(n)$ be the maximal $k$ such that in any set $A\subset \mathbb{R}$ of size $n$ there is a subset $B\subseteq A$ of size $\lvert B\rvert\geq k$ which is dissociated that is, the sums $\sum_{b\in S}b$ are distinct for all $S\subseteq B$. Estimate $f(n)$ - in particular, is it true that\[f(n)\geq \lfloor \log_2 n\rfloor?\]",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/963,
erdos-968,open,needs-theory,mathematics / number theory,"Let $u_n=p_n/n$, where $p_n$ is the $n$th prime. Does the set of $n$ such that $u_n<u_{n+1}$ have positive density?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A387591 and check the conjectured growth against them.,https://www.erdosproblems.com/968,
erdos-969,open,needs-theory,mathematics / number theory,"Let $Q(x)$ count the number of squarefree integers in $[1,x]$. Determine the order of magnitude in the error term in the asymptotic\[Q(x)=\frac{6}{\pi^2}x+E(x).\]",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A013928 and check the conjectured growth against them.,https://www.erdosproblems.com/969,
erdos-97,open,compute-checkable-small-cases,mathematics / geometry,"Open, but could be disproved with a finite counterexample. - $100 Does every convex polygon have a vertex with no other $4$ vertices equidistant from it?","Erdős problems database (status: falsifiable; prize: $100; tags: geometry, distances, convex)",The database marks this as refutable by a finite computation if false: search for a counterexample. Extend OEIS N/A first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/97,
erdos-970,open,needs-theory,mathematics / number theory,"Let $h(k)$ be Jacobsthal's function, defined to as the minimal $m$ such that, if $n$ has at most $k$ prime factors, then in any set of $m$ consecutive integers there exists an integer coprime to $n$. Determine the order of magnitude of $h(k)$. In particular, is it true that\[h(k) \ll k^2?\]",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A048669 and check the conjectured growth against them.,https://www.erdosproblems.com/970,
erdos-971,open,needs-theory,mathematics / number theory,"Let $p(a,d)$ be the least prime congruent to $a\pmod{d}$. Does there exist a constant $c>0$ such that, for all large $d$,\[p(a,d) > (1+c)\phi(d)\log d\]for $\gg \phi(d)$ many values of $a$?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A226521 and check the conjectured growth against them.,https://www.erdosproblems.com/971,
erdos-972,open,needs-theory,mathematics / number theory,Let $\alpha>1$ be irrational. Are there infinitely many primes $p$ such that $\lfloor p\alpha\rfloor$ is also prime?,Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/972,
erdos-973,open,needs-theory,mathematics / analysis,"Does there exist a constant $C>1$ such that, for every $n\geq 2$, there exists a sequence $z_i\in \mathbb{C}$ with $z_1=1$ and $\lvert z_i\rvert \geq 1$ for all $1\leq i\leq n$ with\[\max_{2\leq k\leq n+1}\left\lvert \sum_{1\leq i\leq n}z_i^k\right\rvert < C^{-n}?\]",Erdős problems database (status: open; prize: no; tags: analysis),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/973,
erdos-975,open,needs-theory,mathematics / number theory,"Let $f\in \mathbb{Z}[x]$ be an irreducible non-constant polynomial such that $f(n)\geq 1$ for all large $n\in\mathbb{N}$. Does there exist a constant $c=c(f)>0$ such that\[\sum_{n\leq X} \tau(f(n))\sim cX\log X,\]where $\tau$ is the divisor function?","Erdős problems database (status: open; prize: no; tags: number theory, divisors)","No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A147807, possible and check the conjectured growth against them.",https://www.erdosproblems.com/975,
erdos-976,open,needs-theory,mathematics / number theory,"Let $f\in \mathbb{Z}[x]$ be an irreducible polynomial of degree $d\geq 2$. Let $F_f(n)$ be maximal such that there exists $1\leq m\leq n$ with $f(m)$ is divisible by a prime $\geq F_f(n)$. Equivalently, $F_f(n)$ is the greatest prime divisor of\[\prod_{1\leq m\leq n}f(m).\]Estimate $F_f(n)$. In particular, is it true that $F_f(n)\gg n^{1+c}$ for some constant $c>0$? Or even $\gg n^d$?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/976,
erdos-978,open,needs-theory,mathematics / number theory,"Let $f\in \mathbb{Z}[x]$ be an irreducible polynomial of degree $k>2$ (and suppose that $k\neq 2^l$ for any $l\geq 1$) such that the leading coefficient of $f$ is positive. Does the set of integers $n\geq 1$ for which $f(n)$ is $(k-1)$-power-free have positive density? If $k>3$, and for all primes $p$ there exists $n$ such that $p^{k-2}\nmid f(n)$, then are there infinitely many $n$ for which $f(n)$ is $(k-2)$-power-free? In particular, does\[n^4+2\]represent infinitely many squarefree numbers?",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/978,
erdos-979,open,needs-theory,mathematics / number theory,"Let $k\geq 2$, and let $f_k(n)$ count the number of solutions to\[n=p_1^k+\cdots+p_k^k,\]where the $p_i$ are prime numbers. Is it true that $\limsup f_k(n)=\infty$?",Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A385316, possible and check the conjectured growth against them.",https://www.erdosproblems.com/979,
erdos-98,open,needs-theory,mathematics / geometry,"Let $h(n)$ be such that any $n$ points in $\mathbb{R}^2$, with no three on a line and no four on a circle, determine at least $h(n)$ distinct distances. Does $h(n)/n\to \infty$?","Erdős problems database (status: open; prize: no; tags: geometry, distances)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/98,
erdos-982,open,compute-checkable-small-cases,mathematics / geometry,"Open, but could be disproved with a finite counterexample. If $n$ distinct points in $\mathbb{R}^2$ form a convex polygon then some vertex has at least $\lfloor \frac{n}{2}\rfloor$ different distances to other vertices.","Erdős problems database (status: falsifiable; prize: no; tags: geometry, convex, distances)",The database marks this as refutable by a finite computation if false: search for a counterexample. Extend OEIS A004526 first; a new term is a checkable contribution even without a resolution.,https://www.erdosproblems.com/982,
erdos-983,open,needs-theory,mathematics / number theory,"Let $n\geq 2$ and $\pi(n)<k\leq n$. Let $f(k,n)$ be the smallest integer $r$ such that in any $A\subseteq \{1,\ldots,n\}$ of size $\lvert A\rvert=k$ there exist primes $p_1,\ldots,p_r$ such that $>r$ many $a\in A$ are only divisible by primes from $\{p_1,\ldots,p_r\}$. Is it true that\[2\pi(n^{1/2})-f(\pi(n)+1,n)\to \infty\]as $n\to \infty$? In general, estimate $f(k,n)$, particularly when $\pi(n)+1<k=o(n)$.",Erdős problems database (status: open; prize: no; tags: number theory),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS possible and check the conjectured growth against them.,https://www.erdosproblems.com/983,
erdos-985,open,needs-theory,mathematics / number theory,"Is it true that, for every prime $p$, there is a prime $q<p$ which is a primitive root modulo $p$?",Erdős problems database (status: open; prize: no; tags: number theory),"No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS A002233, A219429, A103309, possible and check the conjectured growth against them.",https://www.erdosproblems.com/985,
erdos-99,open,needs-theory,mathematics / geometry,"Let $A\subseteq\mathbb{R}^2$ be a set of $n$ points with minimum distance equal to 1, chosen to minimise the diameter of $A$. If $n$ is sufficiently large then must there be three points in $A$ which form an equilateral triangle of size 1?","Erdős problems database (status: open; prize: $100; tags: geometry, distances)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/99,
erdos-993,open,compute-checkable-small-cases,mathematics / graph theory,"Open, but could be disproved with a finite counterexample. The independent set sequence of any tree or forest is unimodal. In other words, if $i_k(G)$ counts the number of independent sets of vertices of size $k$ in a graph $G$, and $T$ is any tree or forest, then for some $m\geq 0$ $$i_{0}(T)\leq i_{1}(T)\leq\cdots\leq i_{m}(T)\geq i_{m+1}(T)\geq i_{m+2}(T)\geq\cdots.$$",Erdős problems database (status: falsifiable; prize: no; tags: graph theory),"The database marks this as refutable by a finite computation if false: search for a counterexample. Extend OEIS A000055, possible first; a new term is a checkable contribution even without a resolution.",https://www.erdosproblems.com/993,
erdos-995,open,needs-theory,mathematics / analysis,"Let $n_1<n_2<\cdots$ be a lacunary sequence of integers and $f\in L^2([0,1])$. Estimate the growth of, for almost all $\alpha$,\[\sum_{1\leq k\leq N}f(\{ \alpha n_k\}).\]For example, is it true that, for almost all $\alpha$,\[\sum_{1\leq k\leq N}f(\{ \alpha n_k\})=o(N\sqrt{\log\log N})?\]","Erdős problems database (status: open; prize: no; tags: analysis, discrepancy)",No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/995,
erdos-996,open,needs-theory,mathematics / analysis,"Let $n_1<n_2<\cdots$ be a lacunary sequence of integers, and let $f\in L^2([0,1])$. Let $f_n$ be the $n$th partial sum of the Fourier series of $f(x)$. Is there an absolute constant $C>0$ such that, if\[\| f-f_n\|_2 \ll \frac{1}{(\log\log\log n)^{C}}\]then\[\lim_{N\to\infty}\frac{1}{N}\sum_{k\leq N}f(\{\alpha n_k\})=\int_0^1 f(x)\mathrm{d}x\]for almost every $\alpha$?",Erdős problems database (status: open; prize: no; tags: analysis),No finite decision procedure is known. Cheapest honest step: compute further terms of OEIS N/A and check the conjectured growth against them.,https://www.erdosproblems.com/996,
op-001,open,unclassified,CS / ML agents,What produces the ~2.5-point excess decay in listwise Acc@1 beyond independent per-comparison noise?,falsification of combination,"Fit a correlated-error model (per-candidate bias, e.g. a complexity heuristic) to Zheng Table 3; compare likelihood with the independent model.",https://commons.diy/s/team-science/resources/res_c92a6d1d8185491b8aee60fa9eb2678b,
op-002,open,unclassified,CS / ML agents,"Does Acc@1 at N = 8, 10, 15 on Zheng's ranking subset fall inside the pre-registered band (0.221–0.253, 0.191–0.222, 0.146–0.173)?",falsification of combination,"Re-score the released 18,438-pair corpus at those N with the paper's protocol.",https://commons.diy/s/team-science/resources/res_c92a6d1d8185491b8aee60fa9eb2678b,
op-003,open,unclassified,claim verification / metascience,"Is the contested fraction among multi-evidence claims ~20% in a fourth open-retrieval corpus (HealthVer, COVID-Fact, Check-COVID)?",falsification of combination,Run graph/tests/polarity_concordance.py logic on the corpus; contested fraction and trend-in-k z.,https://commons.diy/s/team-science/resources/res_4a75b957702c4d2a9df534ce202ce607,
op-004,open,unclassified,claim verification / metascience,"Why is contestedness flat in the number of evidence documents? Retrieval surfacing both sides of contested claims, or annotation policy?",open question raised by a finding,Subsample evidence per claim in Climate-FEVER and SciFact-Open and recompute; inspect annotation guidelines for DISPUTED.,https://commons.diy/s/team-science/resources/res_4a75b957702c4d2a9df534ce202ce607,
op-005,open,unclassified,CS / ML agents,Does RPM child selection accuracy in AIRA-dojo track the noisy-argmax curve as the number of children grows?,limitation section of a read paper,Requires an AIRA-dojo run with N children and a frozen judge; compare selection accuracy with the calibrated curve. Not a cheapest test.,https://arxiv.org/abs/2608.13940,
op-006,open,unclassified,biomedical / claim verification,Are SciFact's 47 multi-evidence claims (never re-retrieved in SciFact-Open) contested under open retrieval?,gap noticed during a test,Retrieve those 47 claims against the SciFact-Open 500K corpus with the released retriever; count mixed polarity.,https://commons.diy/s/team-science/resources/res_4a75b957702c4d2a9df534ce202ce607,
op-007,open,unclassified,cross-domain / metascience,"Once corpus size is controlled, does within-claim polarity concordance differ between biomedical and Wikipedia claims?",cross-domain comparison,Match evidence counts across Climate-FEVER and SciFact-Open and compare fitted concordance with bootstrap CIs.,https://commons.diy/s/team-science/resources/res_4a75b957702c4d2a9df534ce202ce607,
op-008,open,unclassified,CS / ML / automated science,"What testable claims does AI-GAs (Clune 2019, arXiv:1905.10985) make? It is cited by 23 ingested papers and carries no claims.",frontier query,"Scout full read, quote-only claims, #177 verdicts.",https://arxiv.org/abs/1905.10985,
op-009,open,unclassified,metascience / scientometrics,"Is Uzzi et al.'s atypical-combination effect visible inside our 2.7k-paper graph, and what would that require (per-paper citation counts)?",method literature,"Needs paper_signal (cited_by_count) rows from OpenAlex, then reference-pair conventionality vs citations.",https://doi.org/10.1126/science.1240474,
op-010,open,unclassified,CS / ML agents,"Do LLM judge errors correlate across candidates that share a surface feature (e.g., code complexity), which would explain op-001?",mechanism hypothesis,Stratify Zheng's pairwise corpus by the complexity heuristic and test whether errors cluster.,https://arxiv.org/abs/2601.05930,
op-011,open,unclassified,mathematics,"Mathematics seed: which entries in the public Erdős problems database (erdosproblems.com) have been resolved since 2020, and can their resolution papers be keyed and ingested as a claims cluster?",human question (operator asked which fields to add),"Scrape the public status list, resolve resolution papers via OpenAlex, ingest as paper rows with one claim each; the database itself is an open-problem source for this table.",https://www.erdosproblems.com/,
op-012,open,unclassified,economics / metascience,Economics seed: do the 18 Camerer et al. 2016 replications of experimental-economics results show the same ~20% contested pattern once original and replication are treated as two evidence documents per claim?,cross-domain comparison (finding 2),Ingest the 18 original + replication pairs as claims with SUPPORTS/REFUTES evidence rows; compute the mixed fraction and compare with the 20% band.,https://doi.org/10.1126/science.aaf0918,
op-013,open,unclassified,physics / cosmology,"Physics seed: is the Hubble-constant tension a contested claim in our registry's sense (both polarities from independent primary measurements), and does the independence baseline mis-predict its persistence?",human question (operator asked which fields to add),Ingest the review and the primary measurement papers it cites; register the claim with per-measurement evidence rows; apply the concordance model.,https://arxiv.org/abs/2103.01183,
op-014,open,unclassified,biomedicine / metascience,"Biology seed beyond claim verification: for the SciFact biomedical claims already in the graph, do the cited abstracts' own reference lists reveal contested subfields when walked one hop (contested fraction by MeSH-like topic)?",gap noticed during a test,"Walk SciFact evidence abstracts one hop with the OpenAlex key, group by OpenAlex topic, compute contested fraction per topic.",https://commons.diy/s/team-science/resources/res_4a75b957702c4d2a9df534ce202ce607,
op-015,open,unclassified,TeamScience ops / metascience,Which field should get a dedicated lens first? Rule under test: add a specialist identity only when a field has ≥10 claims and Scout has bounced off it twice.,human question (operator asked how agents are picked),Track claims per domain and Scout no-ops per field in the explorer; the first field to cross the rule gets a lane identity.,https://commons.diy/s/team-science/resources/res_e4812f82d5f24e9aae97cac4e6533d4a,
se-cstheory-10983,open,compute-checkable-small-cases,theoretical computer science / algorithms,"Optimal algorithm for finding the girth of a sparse graph? I wonder how to find the girth of a sparse undirected graph. By sparse I mean $|E|=O(|V|)$. By optimum I mean the lowest time complexity. I thought about some modification on Tarjan's algorithm for undirected graphs, but I didn't find good results. Actually I thought that if I could find a 2-connected components in $O(|V|)$, then I can find the…","stack exchange open-problems tag (cstheory, score 14, 2 answers)",,https://cstheory.stackexchange.com/questions/10983/optimal-algorithm-for-finding-the-girth-of-a-sparse-graph,
se-cstheory-1160,open,compute-checkable-small-cases,theoretical computer science / general,"Projective Plane of Order 12. Objective : Settle the conjecture that there is no projective plane of order 12. In 1989, using computer search on a Cray, Lam proved that no projective plane of order 10 exists. Now that God's Number for Rubik's Cube has been determined after just a few weeks of massive brute force search (plus clever math of symmetry), it seems to me that this…","stack exchange open-problems tag (cstheory, score 15, 2 answers)",,https://cstheory.stackexchange.com/questions/1160/projective-plane-of-order-12,
se-cstheory-12007,open,needs-theory,theoretical computer science / complexity theory,"Learning with (Signed) Errors. $\underline{\bf Background}$ In 2005, Regev [1] introduced the Learning with Errors (LWE) problem, a generalization of the Learning Parity with Error problem. The assumption of this problem's hardness for certain parameter choices now underlies the security proofs for a host of post-quantum cryptosystems in the field of lattice-based cryptography.…","stack exchange open-problems tag (cstheory, score 10, 1 answers)",,https://cstheory.stackexchange.com/questions/12007/learning-with-signed-errors,
se-cstheory-12351,open,needs-theory,theoretical computer science / complexity theory,Deciding whether an NC${}^0_3$ circuit computes a permutation or not. I would like to ask about a special case of the question “ Deciding if a given NC 0 circuit computes a permutation ” by QiCheng that has been left unanswered. A Boolean circuit is called an NC 0 k circuit if each output gate syntactically depends on at most k input gates. (We say that an output gate g syntactically depends on an input gate g ′…,"stack exchange open-problems tag (cstheory, score 27, 2 answers)",,https://cstheory.stackexchange.com/questions/12351/deciding-whether-an-nc0-3-circuit-computes-a-permutation-or-not,
se-cstheory-1265,open,compute-checkable-small-cases,theoretical computer science / algorithms,"Polynomial time approximation algorithms for machine scheduling: how many open problems are left? In 1999, Petra Schuurman and Gerhard J. Woeginger published the paper ""Polynomial time approximation algorithms for machine scheduling: Ten open problems"" . Since then, to the best of my knowledge, reviews which would concern very the same list of problems haven't appeared. Thus it would be great and useful if each of us could make such summary on…","stack exchange open-problems tag (cstheory, score 22, 2 answers)",,https://cstheory.stackexchange.com/questions/1265/polynomial-time-approximation-algorithms-for-machine-scheduling-how-many-open-p,
se-cstheory-1317,open,needs-theory,theoretical computer science / complexity theory,"Problems between NC and P: How many have been resolved from this list? In the paper ""A Compendium of Problems Complete for P"" by Greenlaw, Hoover and Ruzzo (PS) (PDF) , there is a list of problems in P that are not known to be in NC and not known to be P-complete either. (This list subsumes all the open problems in the excellent survey by Karp and Ramachandran .) The open problems list starts on page 89. How many…","stack exchange open-problems tag (cstheory, score 21, 2 answers)",,https://cstheory.stackexchange.com/questions/1317/problems-between-nc-and-p-how-many-have-been-resolved-from-this-list,
se-cstheory-178,open,compute-checkable-small-cases,theoretical computer science / learning,"Approximating the sign rank of a matrix. The sign rank of a matrix A with +1,-1 entries is the least rank (over the reals) of a matrix B which has the same sign pattern as A (i.e., $A_{ij}B_{ij}>0$ for all $i,j$). This notion is important in communication complexity and learning theory. My question is: Are there any known (subexponential time) algorithms that approximate the sign-rank of…","stack exchange open-problems tag (cstheory, score 25, 2 answers)",,https://cstheory.stackexchange.com/questions/178/approximating-the-sign-rank-of-a-matrix,
se-cstheory-17868,open,needs-theory,theoretical computer science / programming languages,"Research and open challenges in Programming Language Theory. In the spirit of some general discussions like this one, I'm opening this thread with the intention to gather opinions on what are the open challenges and hot topics in research on programming languages . I hope that the discussion might even bring to surface opinions regarding the future of research in programming languages. I believe that this…","stack exchange open-problems tag (cstheory, score 38, 4 answers)",,https://cstheory.stackexchange.com/questions/17868/research-and-open-challenges-in-programming-language-theory,
se-cstheory-1814,open,compute-checkable-small-cases,theoretical computer science / combinatorics,"A combinatorial version for the polynomial Hirsch conjecture. Consider $t$ disjoint families of subsets of {1,2,…,n}, ${\cal F}_1,{\cal F_2},\dots {\cal F_t}$ . Suppose that (*) For every $i \lt j \lt k$ and every $R \in {\cal F}_i$, and $T \in {\cal F}_k$, there is $S \in {\cal F}_j$ which contains $R \cap T$. The basic question is: How large can t be??? What is known The best known upper bound is quasi…","stack exchange open-problems tag (cstheory, score 53, 1 answers)",,https://cstheory.stackexchange.com/questions/1814/a-combinatorial-version-for-the-polynomial-hirsch-conjecture,
se-cstheory-18187,open,compute-checkable-small-cases,theoretical computer science / algebraic complexity,"Identifying Reducible/Irreducible polynomials over $Z[x]$. It is well known LLL algorithm provides a fully polynomial algorithm to factor a reducible primitive polynomial over $\mathbb{Z}[x]$. Say one only seeks to identify whether a given polynomial over $\mathbb{Z}[x]$ is reducible, then what are the best ways known to solve this? If reducible, the algorithm should correctly say yes and if not, it…","stack exchange open-problems tag (cstheory, score 20, 0 answers)",,https://cstheory.stackexchange.com/questions/18187/identifying-reducible-irreducible-polynomials-over-zx,
se-cstheory-1842,open,compute-checkable-small-cases,theoretical computer science / algorithms,"Positive topological ordering, take 3. Suppose we have an n by n matrix. Is it possible to reorder its rows and columns such that we get an upper-triangular matrix? This question is motivated by this problem: Positive topological ordering The original decision problem is at least as hard as this one, so an NP-completeness result would solve that too. Edit: Laszlo Vegh and Andras Frank…","stack exchange open-problems tag (cstheory, score 20, 4 answers)",,https://cstheory.stackexchange.com/questions/1842/positive-topological-ordering-take-3,
se-cstheory-1962,open,compute-checkable-small-cases,theoretical computer science / complexity theory,"Is it NP-hard to play international draughts correctly? Is the following problem NP-hard? Given a board configuration for $n\times n$ international draughts , find a single legal move. The corresponding problem for $n\times n$ American checkers (aka English draughts) is trivially solvable in polynomial time. There are three major differences between these two games. The first and most significant…","stack exchange open-problems tag (cstheory, score 31, 3 answers)",,https://cstheory.stackexchange.com/questions/1962/is-it-np-hard-to-play-international-draughts-correctly,
se-cstheory-2333,open,needs-theory,theoretical computer science / complexity theory,"Any example of an unsatisfiable integer program with non constant Rank Lower bounds for LS+ cuts but with short LS+ refutations? Assume we want to refute an unsatisfiable CNF. We can interpret it as an integer program, thus a refutation can be done by applying Lovasz-Schrijver semidefinite cuts ($LS_{+}$ cuts) to its linear relaxation until the polytope is empty. For a polytope $P$ we consider $N_{+}(P)$ to be the polytope where all possible $LS_{+}$ cuts have been further…","stack exchange open-problems tag (cstheory, score 15, 0 answers)",,https://cstheory.stackexchange.com/questions/2333/any-example-of-an-unsatisfiable-integer-program-with-non-constant-rank-lower-bou,
se-cstheory-25297,open,compute-checkable-small-cases,theoretical computer science / complexity theory,"Is graph coloring complete for poly-APX? Is the graph coloring problem complete for poly-APX under C-reductions (alternatively, under AP-reductions)? For the graph coloring problem, speaking of a feasible solution means a proper coloring for all vertices of the given graph. The complexity class poly-APX contains all NP optimization problems that can be approximated within a factor that…","stack exchange open-problems tag (cstheory, score 16, 0 answers)",,https://cstheory.stackexchange.com/questions/25297/is-graph-coloring-complete-for-poly-apx,
se-cstheory-2559,open,compute-checkable-small-cases,theoretical computer science / complexity theory,"Is $coNP^{\#P}=NP^{\#P}=P^{\#P}$? By http://www.cs.umd.edu/~jkatz/complexity/relativization.pdf If $A$ is a PSPACE-complete language, $P^{A}=NP^{A}$. If $B$ is a deterministic polynomial-time oracle, $P^{B}\ne NP^{B}$ (assuming $P\ne NP$). $PP$ is the class of decision problems analog for $\#P$ and $P\subseteq PP\subseteq PSPACE$, but neither $P=PP$ nor $PP=PSAPCE$ is known. But…","stack exchange open-problems tag (cstheory, score 12, 2 answers)",,https://cstheory.stackexchange.com/questions/2559/is-conp-p-np-p-p-p,
se-cstheory-2562,open,needs-theory,theoretical computer science / algorithms,"One Stack, Two Queues. background Several years ago, when I was an undergraduate, we were given a homework on amortized analysis. I was unable to solve one of the problems. I had asked it in comp.theory , but no satisfactory result came up. I remember the course TA insisted on something he couldn't prove, and said he forgot the proof, and ... [you know what]. Today, I…","stack exchange open-problems tag (cstheory, score 65, 10 answers)",,https://cstheory.stackexchange.com/questions/2562/one-stack-two-queues,
se-cstheory-29317,open,needs-theory,theoretical computer science / complexity theory,"Integer linear programming in logarithmic number of variables. I read that integer linear programming is solvable in polynominal time if the number $n$ of variables is fixed, i.e. $n \in O(1)$. If the number of variables grows logarithmically, i.e. $n \in O(\log_2(N))$ for a given input of size $N$, is the problem still solvable in polynominal time or is this an open","stack exchange open-problems tag (cstheory, score 17, 1 answers)",,https://cstheory.stackexchange.com/questions/29317/integer-linear-programming-in-logarithmic-number-of-variables,
se-cstheory-30591,open,compute-checkable-small-cases,theoretical computer science / algorithms,"Are there any interesting open questions having to do with submodularity, specially in the intersection of theoretical machine learning? I was interested in knowing about open research topics related with sub modularity, specially within its intersection with theoretical machine learning (and related topics). I am particularly interested in algorithms (specially their application to machine learning, and whenever possible, proving theoretical guarantees for them, wether these…","stack exchange open-problems tag (cstheory, score 7, 1 answers)",,https://cstheory.stackexchange.com/questions/30591/are-there-any-interesting-open-questions-having-to-do-with-submodularity-specia,
se-cstheory-3105,open,compute-checkable-small-cases,theoretical computer science / complexity theory,"The complexity of checking whether two DAG have the same number of topological sorts. This problem is highly related to the CNF one. Here is the problem: given two DAG (directed acyclic graphs), if they have the same counting of topological sorts, answer ""Yes"", otherwise, answer ""No"". Intuitively, the complexity of this problem is $C_=P$ -complete, just as the CNF one. It is easy to see it is in $C_=P$, but is it","stack exchange open-problems tag (cstheory, score 31, 0 answers)",,https://cstheory.stackexchange.com/questions/3105/the-complexity-of-checking-whether-two-dag-have-the-same-number-of-topological-s,
se-cstheory-31835,open,compute-checkable-small-cases,theoretical computer science / graph theory,"$\Delta = 57, d=2$ Moore Graph. I am looking into the last open question regarding the existence of Moore Graphs of diameter 2. A problem that has been open in combinatorics for more than 55 years. You may recall that Hoffman and Singleton proved that the only Moore Graphs [: regular graphs achieving the Moore Bound for a given diameter $d$ and a degree $\Delta$] that exist for…","stack exchange open-problems tag (cstheory, score 23, 0 answers)",,https://cstheory.stackexchange.com/questions/31835/delta-57-d-2-moore-graph,
se-cstheory-3309,open,compute-checkable-small-cases,theoretical computer science / formal languages,"Status of Cerny Conjecture? A DFA has a synchronizing word if there is a string that sends any state of the DFA to a single state. In ‘The Cerny Conjecture for Aperiodic Automata” by A. N. Trahtman (Discrete Mathematics and Theoretical Computer Science vol. 9:2, 2007, pp.3-10), he wrote, Cerny conjectured in 1964 that every n-state synchronizable DFA possesses a…","stack exchange open-problems tag (cstheory, score 19, 2 answers)",,https://cstheory.stackexchange.com/questions/3309/status-of-cerny-conjecture,
se-cstheory-33866,open,compute-checkable-small-cases,theoretical computer science / formal languages,"Does there exist a hardest DCFL? Greibach famously defined a language $H$, the so-called nondeterministic version of $D_2$, such that any CFL is an inverse morphic image of $H$. Does there exist a similar statement with DCFL, possibly with some restriction on the morphisms allowed? (See, e.g., M. Autebert, J. Berstel, and L. Boasson. Context-free languages and pushdown automata.…","stack exchange open-problems tag (cstheory, score 12, 3 answers)",,https://cstheory.stackexchange.com/questions/33866/does-there-exist-a-hardest-dcfl,
se-cstheory-34,open,compute-checkable-small-cases,theoretical computer science / algorithms,"How hard is unshuffling a string? A shuffle of two strings is formed by interspersing the characters into a new string, keeping the characters of each string in order. For example, is a shuffle of and . Let me call a string square if it is a shuffle of two identical strings. For example, is square, because it is a shuffle of and , but the string is not square. Is there a fast…","stack exchange open-problems tag (cstheory, score 130, 11 answers)",,https://cstheory.stackexchange.com/questions/34/how-hard-is-unshuffling-a-string,
se-cstheory-34792,open,needs-theory,theoretical computer science / formal languages,"Does PEG contain CFG? Despite their considerable expressive power, all PEG s can be parsed in linear time using a tabular or memoizing parser (8). These properties strongly suggest that CFGs and PEGs define incomparable language classes, although a formal proof that there are context-free languages not expressible via PEGs appears surprisingly elusive. Bryan Ford, ""…","stack exchange open-problems tag (cstheory, score 7, 0 answers)",,https://cstheory.stackexchange.com/questions/34792/does-peg-contain-cfg,
se-cstheory-37384,open,compute-checkable-small-cases,theoretical computer science / np complete,"Does Memcomputing really solve an NP-complete problem? I came across an article published in Science ""Memcomputing NP-complete problems in polynomial time using polynomial resources and collective states"" , which makes some pretty astonishing claims. Memcomputing is a novel non-Turing paradigm of computation that uses interacting memory cells (memprocessors for short) to store and process information…","stack exchange open-problems tag (cstheory, score 10, 4 answers)",,https://cstheory.stackexchange.com/questions/37384/does-memcomputing-really-solve-an-np-complete-problem,
se-cstheory-40062,open,definition-dispute,theoretical computer science / quantum information,Noisy channel coding theorem in quantum information. Why Shannon's noisy channel coding theorem can't be used for quantum communication applications? Schumacher proved the first Noiseless theorem and there are quantum error correction mechanisms out there. I guess I am not sure what is the gap in knowledge to prove the noisy channel coding theorem,"stack exchange open-problems tag (cstheory, score 3, 1 answers)",,https://cstheory.stackexchange.com/questions/40062/noisy-channel-coding-theorem-in-quantum-information,
se-cstheory-4034,open,compute-checkable-small-cases,theoretical computer science / algorithms,"Complexity of computing shortest paths in the plane with polygonal obstacles. Suppose we are given several disjoint simple polygons in the plane, and two points $s$ and $t$ outside every polygon. The Euclidean shortest path problem is to compute the Euclidean shortest path from $s$ to $t$ that does not intersect the interior of any polygon. For concreteness, let us assume that the coordinates of $s$ and $t$, and the…","stack exchange open-problems tag (cstheory, score 23, 1 answers)",,https://cstheory.stackexchange.com/questions/4034/complexity-of-computing-shortest-paths-in-the-plane-with-polygonal-obstacles,
se-cstheory-42654,open,compute-checkable-small-cases,theoretical computer science / formal languages,A conjecture related to the Cerny conjecture - counterexample/reference request. The Cerny conjecture is the statement that any synchronizing automaton with $n$ states has a synchronizing word of length at most $(n-1)^2$ . The best current upper bound for the length of a synchronizing word is $O(n^3)$ . Let's say that two states are merged by a word if that word takes the two states to the same state. A pumping lemma type…,"stack exchange open-problems tag (cstheory, score 8, 1 answers)",,https://cstheory.stackexchange.com/questions/42654/a-conjecture-related-to-the-cerny-conjecture-counterexample-reference-request,
se-cstheory-4301,open,compute-checkable-small-cases,theoretical computer science / formal languages,"Protocol partition number and deterministic communication complexity. Besides (deterministic) communication complexity $cc(R)$ of a relation $R$, another basic measure for the amount of communication needed is the protocol partition number $pp(R)$. The relation between these two measures is known up to a constant factor. The monograph by Kushilevitz and Nisan (1997) gives $$cc(R)/3 \le \log_2(pp(R)) \le cc(R).$$…","stack exchange open-problems tag (cstheory, score 22, 2 answers)",,https://cstheory.stackexchange.com/questions/4301/protocol-partition-number-and-deterministic-communication-complexity,
se-cstheory-4556,open,compute-checkable-small-cases,theoretical computer science / complexity theory,"Space-time tradeoff and the best algorithm. Consider some language $L$ such that: $L \in DTIME(O(f(n))) \cap DSPACE(O(g(n)))$ and so that $L \not\in DTIME(o(f(n))) \cup DSPACE(o(g(n)))$ In other words, the fastest machine $M$ computes $L$ in time $O(f(n))$ and the most space efficient machine $M'$ computes $L$ while using space $O(g(n))$. What can be said about the space efficiency of M or…","stack exchange open-problems tag (cstheory, score 14, 2 answers)",,https://cstheory.stackexchange.com/questions/4556/space-time-tradeoff-and-the-best-algorithm,
se-cstheory-46445,open,compute-checkable-small-cases,theoretical computer science / parameterized complexity,"Open Problems About Nowhere-Dense Classes of Graphs. I'm writing a survey about nowhere-dense graphs. I would like to list some of the main open problems in the field. In particular I would like to list problems of the following form. The problem has been shown to be efficiently solvable (meaning in polynomial time, FPT time, XP time, depending on the context) on the ""largest"" proper sub-classes…","stack exchange open-problems tag (cstheory, score 5, 0 answers)",,https://cstheory.stackexchange.com/questions/46445/open-problems-about-nowhere-dense-classes-of-graphs,
se-cstheory-4882,open,compute-checkable-small-cases,theoretical computer science / complexity theory,"Finding a prime greater than a given bound. Is a deterministic polynomial-time algorithm known for the following problem: Input: a natural number $n$ (in binary encoding) Output: a prime number $p > n$. (According to a list of open problems by Leonard Adleman, the problem was open in","stack exchange open-problems tag (cstheory, score 28, 2 answers)",,https://cstheory.stackexchange.com/questions/4882/finding-a-prime-greater-than-a-given-bound,
se-cstheory-5101,open,compute-checkable-small-cases,theoretical computer science / complexity theory,"Efficiently computable function as a counter-example to Sarnak's Mobius conjecture. Recently, Gil Kalai and Dick Lipton both wrote nice articles on an interesting conjecture proposed by Peter Sarnak, an expert in number theory and the Riemann Hypothesis. Conjecture. Let $\mu(k)$ be the Möbius function . Suppose $f: \mathbb{N} \to \{-1,1\}$ is an $\mathsf{AC}^0$ function with input $k$ in the form of binary representation of $k$ ,…","stack exchange open-problems tag (cstheory, score 37, 1 answers)",,https://cstheory.stackexchange.com/questions/5101/efficiently-computable-function-as-a-counter-example-to-sarnaks-mobius-conjectu,
se-cstheory-515,open,compute-checkable-small-cases,theoretical computer science / complexity theory,Are there alternatives to using polynomials in defining the different notions of efficient computation? Is invoking polynomials in defining the different notions of efficient computation the real obstacle to resolve the P vs NP problem? Do we need a paradigm shift by redefining what constitute an efficient,"stack exchange open-problems tag (cstheory, score 4, 4 answers)",,https://cstheory.stackexchange.com/questions/515/are-there-alternatives-to-using-polynomials-in-defining-the-different-notions-of,
se-cstheory-52968,open,needs-theory,theoretical computer science / graph theory,"Exploding number of homomorphisms. I'm trying to tackle the following problem: given two graphs $A$ and $B$ , if there exists a graph $D$ such that $\hom(A, D) > \hom(B, D)$ (i.e. there is more homomorphisms from $A$ to $D$ than from $B$ to $D$ ), then for any $c > 0$ there is a graph $D_c$ such that $\hom(A, D_c) > c\cdot\hom(B, D_c)$ . The motivation for this comes from the query…","stack exchange open-problems tag (cstheory, score 5, 0 answers)",,https://cstheory.stackexchange.com/questions/52968/exploding-number-of-homomorphisms,
se-cstheory-54043,open,compute-checkable-small-cases,theoretical computer science / complexity theory,"Is there a 'mathematical program' to separate P from BQP? This question has been motivated by the existence of an ongoing (and possibly long-term) program for $P\neq NP$ conjecture like GCT (Mulmuley, 1999). Usually, such programs are marked by long and short-term goals. (see Fortnow review article ) A 'decent' number of people recognise the program's viability. Is there a similar program for $P$ vs…","stack exchange open-problems tag (cstheory, score 4, 0 answers)",,https://cstheory.stackexchange.com/questions/54043/is-there-a-mathematical-program-to-separate-p-from-bqp,
se-cstheory-55773,open,compute-checkable-small-cases,theoretical computer science / automata theory,"Efficiently finding an infix of a word that is rejected by an NFA. I am interested in the fine-grained complexity of the following NFA infix rejection problem: Input: a word $w$ , a nondeterministic finite automaton $A$ Output: is there an infix of $w$ (i.e., some $u$ with $w = sut$ for some $s,t$ ) that is rejected by $A$ The naive algorithm for NFA infix rejection is to consider every infix of $w$ and run a…","stack exchange open-problems tag (cstheory, score 4, 2 answers)",,https://cstheory.stackexchange.com/questions/55773/efficiently-finding-an-infix-of-a-word-that-is-rejected-by-an-nfa,
se-cstheory-6430,open,compute-checkable-small-cases,theoretical computer science / quantum computing,"Quantum PAC learning. Background Functions in $AC^0$ are PAC learnable in quasipolynomial time with a classical algorithm that requires $O(2^{log(n)^{O(d)}})$ randomly chosen queries to learn a circuit of depth d [1]. If there is no $2^{n^{o(1)}}$ factoring algorithm then this this is optimal [2]. Of course, on a quantum computer we know how to factor, so this lower…","stack exchange open-problems tag (cstheory, score 18, 2 answers)",,https://cstheory.stackexchange.com/questions/6430/quantum-pac-learning,
se-cstheory-783,open,compute-checkable-small-cases,theoretical computer science / complexity theory,"Is optimally solving the n×n×n Rubik's Cube NP-hard? Consider the obvious $n\times n\times n$ generalization of the Rubik's Cube . Is it NP-hard to compute the shortest sequence of moves that solves a given scrambled cube, or is there a polynomial-time algorithm? [Some related results are described in my recent blog post","stack exchange open-problems tag (cstheory, score 41, 3 answers)",,https://cstheory.stackexchange.com/questions/783/is-optimally-solving-the-n%c3%97n%c3%97n-rubiks-cube-np-hard,
se-cstheory-791,open,compute-checkable-small-cases,theoretical computer science / combinatorics,"Grid $k$-coloring without monochromatic rectangles. Update : The obstruction set (i.e. the NxM ""barrier"" between colorable and uncolorable grid sizes) for all monochromatic-rectangle-free 4-colorings is now known . Anyone feel up to trying 5-colorings? ;) The following question arises out of Ramsey Theory . Consider a $k$-coloring of the $n$-by-$m$ grid graph. A exists whenever four cells with the…","stack exchange open-problems tag (cstheory, score 38, 6 answers)",,https://cstheory.stackexchange.com/questions/791/grid-k-coloring-without-monochromatic-rectangles,
se-cstheory-8047,open,compute-checkable-small-cases,theoretical computer science / complexity theory,"Complexity of balanced graph partition problem. Wagner and Wagner, in "" Between min cut and graph bisection "" (MFCS 1993), studied a variant of minimum bisection problem where we seek a cut with minimum size such that each partition has at least $\log n$ verticies. They stated that the complexity of this variant is an open problem since they did not find an efficient algorithm nor NP-hardness…","stack exchange open-problems tag (cstheory, score 5, 0 answers)",,https://cstheory.stackexchange.com/questions/8047/complexity-of-balanced-graph-partition-problem,
se-cstheory-8093,open,compute-checkable-small-cases,theoretical computer science / algorithms,"Given a graph, decide if its edge connectivity is at least n/2 or not. Chapter 1 of the book The Probabilistic Method, by Alon and Spencer mentions the following problem: Given a graph $G$, decide if its edge connectivity is at least $n/2$ or not. The author mentions the existence of a $O(n^3)$ algorithm by Matula and improves it to $O(n^{8/3}\log n)$. My question is, what's the best known running time for this…","stack exchange open-problems tag (cstheory, score 13, 1 answers)",,https://cstheory.stackexchange.com/questions/8093/given-a-graph-decide-if-its-edge-connectivity-is-at-least-n-2-or-not,
se-mo-100816,open,compute-checkable-small-cases,mathematics / graph theory,"Maximal class of simple graphs of order $n$ with mutually distinct numbers of spanning trees. This problem in some ways related to this post. Let $A_n$ be the set of all integers $x$ such that there exist a connected simple graph of order $n$ having precisely $x$ spanning trees. Study the growth rate of $|A_n|.$ The question was raised by J.Sedlacek in his paper entitled: On the number of spanning trees of finite graphs, Cas. Pro. Pest…","stack exchange open-problems tag (mathoverflow, score 8, 1 answers)",,https://mathoverflow.net/questions/100816/maximal-class-of-simple-graphs-of-order-n-with-mutually-distinct-numbers-of-sp,
se-mo-10223,open,compute-checkable-small-cases,mathematics / number theory,"Linear equation with primes. Is there an integer $n$ with an infinite number of representations of the form $n=2q-p$, where $p$ and $q$ are both primes? Given a positive integer $k>1$, I would like to know for which (if any) integers $n$ the linear equation $q-kp=n$ admits an infinite number of solutions, where $p$ and $q$ are primes. (I'm not including $k=1$ because it…","stack exchange open-problems tag (mathoverflow, score 10, 1 answers)",,https://mathoverflow.net/questions/10223/linear-equation-with-primes,
se-mo-102644,open,compute-checkable-small-cases,mathematics / group actions,"Actions on ℍⁿ generated by torsion elements. Let $n$ be a large integer. I am looking for a cocompact properly discontinuous isometric action on $n$-dimensional Lobachevky space which is generated by elements of finite order. Or equivalently, I need a cocompact properly discontinuous isometric action $\Gamma\curvearrowright\mathbb{H}^n$ such that $\mathbb{H}^n/\Gamma$ is simply connected,…","stack exchange open-problems tag (mathoverflow, score 17, 0 answers)",,https://mathoverflow.net/questions/102644/actions-on-%e2%84%8d%e2%81%bf-generated-by-torsion-elements,
se-mo-10356,open,needs-theory,mathematics / functional analysis,"Topological ""Interpolation"" ? Let E be a normed space, and let $T$: E * $\rightarrow$ E * be a nonlinear operator. Suppose that : 1) $T$ is continuous from ( E * , ||.||) to itself (i.e., it is norm -continuous). and 2) $T$ is continuous from ( E * , w * ) to itself (i.e., it is weakly-star continuous). Then does it follow that 3) $T$ is continuous from ( E * , w ) to itself…","stack exchange open-problems tag (mathoverflow, score 9, 1 answers)",,https://mathoverflow.net/questions/10356/topological-interpolation,
se-mo-10358,open,compute-checkable-small-cases,mathematics / optimization and control,"Solving NP problems in (usually) Polynomial time? Just because a problem is NP-complete doesn't mean it can't be usually solved quickly. The best example of this is probably the traveling salesman problem, for which extraordinarily large instances have been optimally solved using advanced heuristics, for instance sophisticated variations of branch-and-bound . The size of problems that can be…","stack exchange open-problems tag (mathoverflow, score 31, 7 answers)",,https://mathoverflow.net/questions/10358/solving-np-problems-in-usually-polynomial-time,
se-mo-104326,open,needs-theory,mathematics / geometric topology,"Motivation and unsolved problems of TQFT. I have been studying topological quantum field theory by mainly reading the Turaev's book. I'd like to know if there are unsolved problems that motivate mathematicians to study TQFT, like Riemann's hypothesis for number theory. I also would like to know if there is a paper or book that list big or small unsolved problems of TQFT. If not, could you…","stack exchange open-problems tag (mathoverflow, score 15, 3 answers)",,https://mathoverflow.net/questions/104326/motivation-and-unsolved-problems-of-tqft,
se-mo-10449,open,needs-theory,mathematics / real algebra,"Artin-Schreier Theorem for Rings. This has been in my mind for quite some time. Looking at the Artin-Schreier Theorem for fields: If $L$ is a field and $K$ its algebraic closure and if $1< [K:L] < \infty$ then $K=L[i]$ and $L$ is a real closed field. (Thus $L$ has characteristic 0. Here $i$ is just the square root of $-1$ ). I was wondering if a ""generalized"" Artin-Schreier exists…","stack exchange open-problems tag (mathoverflow, score 7, 0 answers)",,https://mathoverflow.net/questions/10449/artin-schreier-theorem-for-rings,
se-mo-104573,open,needs-theory,mathematics / riemannian geometry,"Open problems in sub-Riemannian geometry. What are some open problems in sub-Riemannian geometry ? I am interested especially in problems concerning connections and curvature, but any contribution is","stack exchange open-problems tag (mathoverflow, score 7, 3 answers)",,https://mathoverflow.net/questions/104573/open-problems-in-sub-riemannian-geometry,
se-mo-105065,open,needs-theory,mathematics / metric geometry,"Minimizing the excursion of a sum of unit vectors. I have $n$ unit-length vectors $v_i$ in $\mathbb{R}^3$ , whose sum is zero: $$ v_1 + v_2 + \cdots + v_n = 0 \; .$$ Now I form the closed polygon $P$ in space by placing them head to tail. So the vertices of $P$ are $$ 0, v_1, (v_1+v_2), \ldots, (v_1+\cdots +v_{n-1}), 0 \; .$$ My question is: What is the minimum excursion from the origin achievable…","stack exchange open-problems tag (mathoverflow, score 17, 1 answers)",,https://mathoverflow.net/questions/105065/minimizing-the-excursion-of-a-sum-of-unit-vectors,
se-mo-1051,open,needs-theory,mathematics / number theory,"Splitting Pythagorean triples. Can one partition the set of positive integers into finitely many Pythagorean-triple-free subsets? If so, what is the smallest number of such subsets? Taking a wild guess, I would be least surprised if the answer were 3. Notice that the 2 subsets of integers such that highest power of 5 that divides them is a) even b) odd manage to split most…","stack exchange open-problems tag (mathoverflow, score 21, 4 answers)",,https://mathoverflow.net/questions/1051/splitting-pythagorean-triples,
se-mo-105722,open,compute-checkable-small-cases,mathematics / matrix theory,S-matrix conjecture: status? Is the $S$-matrix conjecture still open? I mean the one listed as Problem 7 in this survey,"stack exchange open-problems tag (mathoverflow, score 3, 2 answers)",,https://mathoverflow.net/questions/105722/s-matrix-conjecture-status,
se-mo-106660,open,needs-theory,mathematics / number theory,"Kronecker's Jugendtraum for real quadratic fields? Kronecker's Jugendtraum (or Hilbert's 12'th problem) is to find abelian extensions of arbitrary number fields by adjoining `special' values of transcendental functions. The Kronecker-Weber theorem was the first realisation of this: i.e. $\mathbb{Q}^{ab}=\mathbb{Q}^{cycl}=\mathbb{Q}(e^{2\pi i \mathbb{Q}})$. If $K$ is an imaginary quadratic field,…","stack exchange open-problems tag (mathoverflow, score 13, 3 answers)",,https://mathoverflow.net/questions/106660/kroneckers-jugendtraum-for-real-quadratic-fields,
se-mo-106815,open,needs-theory,mathematics / classical analysis and odes,"Measures of full Hausdorff dimension for self-affine sets. Consider the iterated function system $T_{1}(x)=(\beta x,\tau y)$, $T_{2}(x,y)=(\beta x+(1-\beta),\tau y+ (1-\tau))$ for $\beta\in(1/2,1)$ and $\tau\in (0,1/2)$ with self affine set $\Lambda_{\beta,\tau}$. It is known that for almost all $\beta$ and all $\tau$ $\dim_{H}\Lambda_{\beta,\tau}=\dim b=1-\log(2\beta)/\log(\tau)$ where $b=(1/2,1/2)$ is…","stack exchange open-problems tag (mathoverflow, score 6, 2 answers)",,https://mathoverflow.net/questions/106815/measures-of-full-hausdorff-dimension-for-self-affine-sets,
se-mo-10752,open,needs-theory,mathematics / combinatorics,"Enumerating (generalized) de Bruijn tori. Given a cyclic word $w$ of length $N$ over a $q$-ary alphabet and $k \in \mathbb{Z}_+$, consider the directed multigraph $G_k(w) = (V,E)$ with $V \subset$ {$1,\dots,q$}$^k$ given by the $k$-lets (i.e., subwords of $k$ symbols) that appear in $w$ (without multiplicity) and $E$ given by the $(k+1)$-lets in $w$ that appear with multiplicity. An edge…","stack exchange open-problems tag (mathoverflow, score 6, 0 answers)",,https://mathoverflow.net/questions/10752/enumerating-generalized-de-bruijn-tori,
se-mo-107534,open,needs-theory,mathematics / real analysis,"Is $Q_n(x)=\sigma_{n+1}(x)/\sigma_n(x)$ logarithmically convex on $\mathbf{R}$? In 1975 J. van de Lune considered the monotony properties of the canonical Riemann Upper and Lower sums for $\int_0^1 t^xdt$, with $x>0$. Writing $\sigma_n(x) := 1^x+2^x+\cdots+n^x$ these sums are $$U_n:=U_n(x)=\sigma_n(x)/n^{x+1} \quad \text{and}\quad L_n:= L_n(x)=\sigma_{n-1}(x)/n^{x+1}.$$ He proved (by mathematical induction) that $U_n>U_{n+1}$…","stack exchange open-problems tag (mathoverflow, score 10, 1 answers)",,https://mathoverflow.net/questions/107534/is-q-nx-sigma-n1x-sigma-nx-logarithmically-convex-on-mathbfr,
se-mo-10799,open,needs-theory,mathematics / combinatorics,"Optimal monotone families for the discrete isoperimetric inequality. Background: the discrete isoperimetric inequality Start with a set $X=\{1,2,...,n\}$ of $n$ elements and the family $2^X$ of all subsets of $X$ . For a real number $p$ between zero and one, we consider a probability distribution $\mu_p$ on $2^X$ where the probability that $i \in S$ is $p$ , independently for different $i$ 's. Thus for $p=1/2$ we…","stack exchange open-problems tag (mathoverflow, score 17, 1 answers)",,https://mathoverflow.net/questions/10799/optimal-monotone-families-for-the-discrete-isoperimetric-inequality,
se-mo-108000,open,compute-checkable-small-cases,mathematics / number theory,"Are Galois groups of Q with restricted ramification supposed to be finitely generated? Fix a finite set $S$ of places of $\mathbb Q$. Let $G_{\mathbb Q,S}$ be the Galois group of the maximal extension of $\mathbb Q$ unramified outside S$. I believe that it is an open question whether this group is topologically finitely generated, that is, contains a dense, finitely generated subgroup. Is there a standard conjecture about whether…","stack exchange open-problems tag (mathoverflow, score 12, 1 answers)",,https://mathoverflow.net/questions/108000/are-galois-groups-of-q-with-restricted-ramification-supposed-to-be-finitely-gene,
se-mo-108404,open,compute-checkable-small-cases,mathematics / hopf algebras,Kaplansky's 6th conjecture: dim(Irrep) | dim(algebra) - for semi-simple Hopf algebras. Let $H$ be a semisimple Hopf algebra. One of the Kaplansky's conjectures states that the dimension of any irreducible $H$-module divides the dimension of $H$. In which cases the conjecture is known to be,"stack exchange open-problems tag (mathoverflow, score 16, 3 answers)",,https://mathoverflow.net/questions/108404/kaplanskys-6th-conjecture-dimirrep-dimalgebra-for-semi-simple-hopf-alg,
se-mo-108406,open,compute-checkable-small-cases,mathematics / representation theory,"Why would dim primitive irrep divide size of some conjugacy class ? From Isaacs et.al. 2005 Conjecture C. Let χ be a primitive irreducible character of an arbitrary finite group G. Then χ(1) divides | clG(g)| for some element g ∈ G. Here, of course, we have written clG(g) to denote the class of g in G. We have checked that Conjecture C holds for all irreducible characters (primitive or not) of all groups in the…","stack exchange open-problems tag (mathoverflow, score 12, 1 answers)",,https://mathoverflow.net/questions/108406/why-would-dim-primitive-irrep-divide-size-of-some-conjugacy-class,
se-mo-108530,open,needs-theory,mathematics / matrices,"A maximal element, where Schur gives a minimal element. Let me recall a result due to I. Schur, which I learnt from F. Goldberg's answer to my MO question Hadamard-like inequalites for positive definite symmetric matrices . If $H$ is a subgroup of $\frak S_n$ and $\chi$ is an irreducible complex character over $H$, define $$d_\chi(S)=\frac1{\chi(e)}\sum_{g\in H}\chi(g)\prod_{i=1}^ns_{ig(i)}.$$ Then for…","stack exchange open-problems tag (mathoverflow, score 13, 1 answers)",,https://mathoverflow.net/questions/108530/a-maximal-element-where-schur-gives-a-minimal-element,
se-mo-108661,open,compute-checkable-small-cases,mathematics / harmonic analysis,"approximate uncertainty principle for finite abelian groups. Edit: we cannot find such an example. It would imply a negative solution to the KS${}_2$ conjecture which has now been proven by Marcus, Spielman, and Srivastava in this paper . In fact, their solution implies that there always exists $S$ such that any $f$ supported on $S$ (or $S^c$) satisfies $\|\frac{1}{2}\hat{f} - \hat{f}|_T\|_2 \leq…","stack exchange open-problems tag (mathoverflow, score 8, 1 answers)",,https://mathoverflow.net/questions/108661/approximate-uncertainty-principle-for-finite-abelian-groups,
se-mo-109045,open,compute-checkable-small-cases,mathematics / representation theory,"If all real conjugacy classes are strongly real, then all real irreps are ""strongly real""(symmetric), true? Question Is true that if all real conjugacy classes of a finite group are strongly real, then all its real irreducible representations (irreps) are ""strongly real"" (symmetric)? And vice versa ? Definitions: An irrep is called real if its Frobenius–Schur indicator is either $+1$ or $-1$ , but not $0$ . An irrep is called ""symmetric"" (""orthogonal""…","stack exchange open-problems tag (mathoverflow, score 4, 2 answers)",,https://mathoverflow.net/questions/109045/if-all-real-conjugacy-classes-are-strongly-real-then-all-real-irreps-are-stron,
se-mo-11091,open,compute-checkable-small-cases,mathematics / group theory,"А generalization of Gromov's theorem on polynomial growth. I was sure it is known, but it appears to be an open problem (see the answer of Terry Tao). Assume for a group $G$ there is a polynomial $P$ such that given $n\in\mathbb N$ there is set of generators $S=S^{-1}$ such that $$|S^n|\leqslant P(n)\cdot|S|\ \ (\text{or stronger condition}\ |S^k|\leqslant P(k)\cdot|S|\ \text{for all}\ k\le n) .$$ Then…","stack exchange open-problems tag (mathoverflow, score 20, 1 answers)",,https://mathoverflow.net/questions/11091/%d0%90-generalization-of-gromovs-theorem-on-polynomial-growth,
se-mo-112417,open,compute-checkable-small-cases,mathematics / symplectic geometry,"Why Donaldson's Four-Six Conjecture? Simon Donaldson apparently made the following conjecture: Two closed symplectic 4-manifolds $(X_1,\omega_1)$ and $(X_2,\omega_2)$ are diffeomorphic if and only if $(X_1\times S^2,\omega_1\oplus\omega)$ is deformation-equivalent to $(X_2\times S^2,\omega_2\oplus\omega)$ . Here $\omega$ is a symplectic structure on $S^2$ , and a…","stack exchange open-problems tag (mathoverflow, score 17, 2 answers)",,https://mathoverflow.net/questions/112417/why-donaldsons-four-six-conjecture,
se-mo-11316,open,compute-checkable-small-cases,mathematics / number theory,"Smooth proper schemes over Z with points everywhere locally. This is a variation on Poonen's question , taking Buzzard's fabulous example into account. It was earlier a part of this other question . Question. Is there a smooth proper scheme $X\to\operatorname{Spec}(\mathbb{Z})$ such that $X(\mathbb{Q}_v)\neq\emptyset$ for every place $v$ of $\ \mathbb{Q}$ ( including $v=\infty$), and yet…","stack exchange open-problems tag (mathoverflow, score 24, 0 answers)",,https://mathoverflow.net/questions/11316/smooth-proper-schemes-over-z-with-points-everywhere-locally,
se-mo-11349,open,needs-theory,mathematics / elliptic curves,Are most cubic plane curves over the rationals elliptic? %This is a new version of the original question modified in the light of the answers and comments. The word 'most' in the title is ambiguous. The following is one way of making it precise. Question1: (This seems to be open. See Poonen's answer below) A cubic projective curve over $\mathbb{Q}$ is given by ten relatively prime integers (the…,"stack exchange open-problems tag (mathoverflow, score 29, 4 answers)",,https://mathoverflow.net/questions/11349/are-most-cubic-plane-curves-over-the-rationals-elliptic,
se-mo-113959,open,compute-checkable-small-cases,mathematics / number theory,"A function whose fixed points are the primes. If $a(n) = (\text{largest proper divisor of } n)$, let $f:\mathbb{N} \setminus \{ 0,1\} \to \mathbb{N}$ be defined by $f(n) = n+a(n)-1$. For instance, $f(100)=100+50-1=149$. Clearly the fixed points of $f$ are the primes. Is every number preperiodic? In other words, is $f(f(\ldots(f(n)\ldots))$ eventually","stack exchange open-problems tag (mathoverflow, score 48, 1 answers)",,https://mathoverflow.net/questions/113959/a-function-whose-fixed-points-are-the-primes,
se-mo-114646,open,needs-theory,mathematics / combinatorics,Intersecting family of triangulations. Let $\cal T_n$ be the family of all triangulations on an $n$ -gon using $(n-3)$ non-intersecting diagonals. The number of triangulations in $\cal T_n$ is $C_{n-2}$ the $(n-2)$ th Catalan number . Let $\cal S \subset \cal T_n$ be a subfamily of triangulations with the property that every two triangulations of $\cal S$ have a common diagonal.…,"stack exchange open-problems tag (mathoverflow, score 57, 1 answers)",,https://mathoverflow.net/questions/114646/intersecting-family-of-triangulations,
se-mo-114758,open,needs-theory,mathematics / real analysis,"Transcendentality of all irrationals in the Cantor set. Hi, I am a student researcher trying to prove that all irrationals within the Cantor set are transcendental. This is grounded, intuitively, in Cantor set members' being non-normal; since algebraic numbers are widely believed to be normal, this implies the transcendentality of the irrationals in the Cantor set. Now, I am at a dead end in trying to…","stack exchange open-problems tag (mathoverflow, score 12, 1 answers)",,https://mathoverflow.net/questions/114758/transcendentality-of-all-irrationals-in-the-cantor-set,
se-mo-115001,open,needs-theory,mathematics / algebraic geometry,"Open problems in Birational Geometry, after BCHM. Rencently a breakthrough was made in the context of the Minimal Model Program by the work of Birkar-Cascini-Hacon-McKernan. They proved that the canonical ring of a smooth or mildly singular projective algebraic variety is finitely generated. Since I'm a master student and so I have no a wide view of the subject (I'm not an expert), I would like…","stack exchange open-problems tag (mathoverflow, score 35, 4 answers)",,https://mathoverflow.net/questions/115001/open-problems-in-birational-geometry-after-bchm,
se-mo-11675,open,compute-checkable-small-cases,mathematics / functional analysis,"Borsuk pairs of Banach spaces. Given $X$, $Y$ two real Banach spaces, let's say that $(X,\ Y)$ is a Borsuk pair if for any continuous mapping $T$ : {$x$ $\in$ $X$ ; $||x||\leq1$} $\rightarrow$ $Y$ s.t. $T$ is odd on {$x$ $\in$ $X$ ; $||x||=1$}, it follows that the set $T^{-1}$( { 0 } ) is nonempty. My conjecture is : $(X,\ Y)$ Borsuk pair $\Longleftrightarrow$ dim $Y$ <…","stack exchange open-problems tag (mathoverflow, score 9, 2 answers)",,https://mathoverflow.net/questions/11675/borsuk-pairs-of-banach-spaces,
se-mo-116947,open,needs-theory,mathematics / classical analysis and odes,"Center of mass from the abstract point of view, or could the ancient Greeks invent modern analysis? This is a very open-ended question, which may or may not have a perfect answer, and for which I have a few ideas but nothing like a clear picture. However, I guess it won't hurt to ask to see if people think about such things at all and if they do, what their ideas are. I don't know whether to make it CW or not: on one hand, it is pure…","stack exchange open-problems tag (mathoverflow, score 20, 2 answers)",,https://mathoverflow.net/questions/116947/center-of-mass-from-the-abstract-point-of-view-or-could-the-ancient-greeks-inve,
se-mo-118545,open,needs-theory,mathematics / combinatorial designs,"What are the major open problems in design theory nowaday? I gather that the question whether the Bruck-Chowla-Ryser condition was sufficient used to top the list, but now that that's settled - what is considered the most interesting open","stack exchange open-problems tag (mathoverflow, score 13, 4 answers)",,https://mathoverflow.net/questions/118545/what-are-the-major-open-problems-in-design-theory-nowaday,
se-mo-119352,open,needs-theory,mathematics / combinatorics,"Randomly switching street lights, in a square city. This is a combinatorics-probability question, best stated however in ""recreational"" terms. Imagine a $N\times N$ city, meaning that we have $N$ horizontal streets, and $N$ vertical streets. At each crossroads there is a street light. When evening comes, some of the lights are switched on, namely those corresponding to a certain given subset…","stack exchange open-problems tag (mathoverflow, score 17, 1 answers)",,https://mathoverflow.net/questions/119352/randomly-switching-street-lights-in-a-square-city,
se-mo-120175,open,compute-checkable-small-cases,mathematics / number theory,"How many values a polynomial map misses? Let $F$ be a field. For a uni-variate polynomial $f(x)$ over $F$,let $M_f(F)$ denote the number of values that $f$ misses, that is, the cardinality of the subset $F - f(F)$ in $F$. Assume that $f$ is not surjective, that is, $M_f(F) > 0$. Is the number $M_f(F)$ big? If $F = {\mathbb F}_q$ is a finite field of $q$ elements and $f\in…","stack exchange open-problems tag (mathoverflow, score 7, 0 answers)",,https://mathoverflow.net/questions/120175/how-many-values-a-polynomial-map-misses,
se-mo-120819,open,needs-theory,mathematics / differential geometry,"open problems in Seiberg-Witten Theory on 4-Manifolds. What are some of the open problems in Seiberg-Witten Theory on 4-Manifolds.I tried googling but couldn't any. I tried googling it, but couldn't find any resources.The places where I can a survey or review of them would be","stack exchange open-problems tag (mathoverflow, score 16, 3 answers)",,https://mathoverflow.net/questions/120819/open-problems-in-seiberg-witten-theory-on-4-manifolds,
se-mo-121010,open,compute-checkable-small-cases,mathematics / logic,"Is there any o-minimal expansion of the real field with functions of growth higher than exponential? Let $\bar{\mathbb{R}}$ be the structure of the real field, that is $(\mathbb{R},0,1,+,-,*,<)$ . We say that a function $f$ is of growth higher than exponential if for all $N\in \mathbb{N}$ there $f(x)$ is ultimately greater than $\exp^N(x)$. That is there is we can find $r\in \mathbb{R}$ such that for all $x>k$ we have $\exp^N(x) < f(x)$ where…","stack exchange open-problems tag (mathoverflow, score 14, 1 answers)",,https://mathoverflow.net/questions/121010/is-there-any-o-minimal-expansion-of-the-real-field-with-functions-of-growth-high,
se-mo-122199,open,compute-checkable-small-cases,mathematics / group theory,"Is it true that every f.g. infinite simple group has exponential growth? Is it true that every finitely generated infinite simple group has exponential (word-)growth? Remark: As Mark Sapir has pointed out, the question whether every finitely generated group of subexponential growth is even residually finite has been answered in the negative in Anna Erschler. Not residually finite groups of intermediate growth,…","stack exchange open-problems tag (mathoverflow, score 16, 1 answers)",,https://mathoverflow.net/questions/122199/is-it-true-that-every-f-g-infinite-simple-group-has-exponential-growth,
se-mo-12284,open,needs-theory,mathematics / algebraic geometry,"Topologically distinct Calabi-Yau threefolds. In dimensions 1 and 2 there is only one, respectively 2, compact Kaehler manifolds with zero first Chern class, up to diffeomorphism. However, it is an open problem whether or not the number of topological types of such manifolds of dimension 3 (Calabi-Yau threefolds) is bounded. I would like to ask what is known in this direction. In particular,…","stack exchange open-problems tag (mathoverflow, score 29, 3 answers)",,https://mathoverflow.net/questions/12284/topologically-distinct-calabi-yau-threefolds,
se-mo-127530,open,needs-theory,mathematics / differential geometry,"Surfaces filled densely by a geodesic. Which smooth, closed surfaces $S \subset \mathbb{R}^3$ have no single geodesic $\gamma$ that fills $S$ densely? Say a geodesic $\gamma$ ""fills $S$ densely"" if the closure of the set of points through which $\gamma$ passes equals $S$. Some examples: A sphere: every geodesic is a great circle. Zoll surfaces, as discussed here: "" Surfaces all of…","stack exchange open-problems tag (mathoverflow, score 44, 5 answers)",,https://mathoverflow.net/questions/127530/surfaces-filled-densely-by-a-geodesic,
se-mo-12758,open,needs-theory,mathematics / algebraic geometry,"Deligne-Simpson problem in the symmetric group. Question. Let $C_1,\dots,C_k$ be conjugacy classes in the symmetric group $S_n$. (More explicitly, each $C_i$ is given by a partition of $n$; $C_i$ consists of permutations whose cycles have the length prescribed by the partition.) Give a necessary and sufficient condition on $C_i$ that would ensure that there are permutations $\sigma_i\in C_i$…","stack exchange open-problems tag (mathoverflow, score 20, 2 answers)",,https://mathoverflow.net/questions/12758/deligne-simpson-problem-in-the-symmetric-group,
se-mo-131973,open,compute-checkable-small-cases,mathematics / number theory,"Distinctive property of the primes 17 and 19? Consider the question whether it is true that a prime number $p$ divides $1^1+2^2+3^3+....+(p-1)^{p-1}$ if and only if $p \in \{17,19\}$. For the obvious heuristic reasons, for large $n$ one would expect there to be roughly $\ln(\ln(n))$ such primes $p < n$, however it seems that presently no examples other than 17 and 19 are known. Is there a…","stack exchange open-problems tag (mathoverflow, score 11, 2 answers)",,https://mathoverflow.net/questions/131973/distinctive-property-of-the-primes-17-and-19,
se-mo-131998,open,needs-theory,mathematics / metric geometry,"Needle probing for a convex body. Suppose there is an unknown closed convex body $K$ of volume vol$(K) = V$ inside the unit cube $[-\frac{1}{2}, \frac{1}{2}]^d$ in $\mathbb{R}^d$. You are permitted to probe with a (one-dimensional) ray $r$, which detects whether $r$ includes some point of $K$, or instead if $r \cap K = \varnothing$. My question is: Q1 . What is the fewest number…","stack exchange open-problems tag (mathoverflow, score 11, 1 answers)",,https://mathoverflow.net/questions/131998/needle-probing-for-a-convex-body,
se-mo-133771,open,needs-theory,mathematics / algebraic geometry,"Does this modified Hasse principle hold for curves? Let $C$ be a curve over $\mathbb Q$ with a point $P$ on $Pic^1$. For each $\mathbb Q$-rational point $Q$, $Q-P$ is a point on the Jacobian $J$. We can use the map $H^0(\mathbb Q, J) \to H^1(\mathbb Q,J[n])$ to obtain a class in $H^1(\mathbb Q,J[n])$ associated to each rational point. Rather than trying to find a local obstruction to existence of…","stack exchange open-problems tag (mathoverflow, score 4, 2 answers)",,https://mathoverflow.net/questions/133771/does-this-modified-hasse-principle-hold-for-curves,
se-mo-13394,open,compute-checkable-small-cases,mathematics / functional analysis,"Finite Rank Commutators. My former student Detelin Dosev and I are interested in classifying the commutators in $L(X)$, the bounded linear operators on the Banach space $X$ (see our joint paper on my home page or the ArXiv and the references therein, and recall that $T$ is a commutator means that there are $A$ and $B$ in $L(X)$ so that $T=AB-BA$). If $T$ is a finite rank…","stack exchange open-problems tag (mathoverflow, score 16, 0 answers)",,https://mathoverflow.net/questions/13394/finite-rank-commutators,
se-mo-13437,open,needs-theory,mathematics / functional analysis,"Hilbert subspaces of indefinite inner product spaces. Let $E$ be a real linear space, endowed with a non-degenerate symmetric bilinear form $(.,.)$. Suppose that the [indefinite] inner product space $(E,(.,.))$ satisfies the following [sequential] properties: ( WSC ) If the sequence { ${(x_{n},\ y)}$ } is Cauchy for each $y$ in $E$, then there exists some $x$ in $E$ such that $\left(x_{n}-x,y\right)$…","stack exchange open-problems tag (mathoverflow, score 6, 0 answers)",,https://mathoverflow.net/questions/13437/hilbert-subspaces-of-indefinite-inner-product-spaces,
se-mo-13444,open,compute-checkable-small-cases,mathematics / number theory,"Prime divisors of numbers 2^n + 3. I'm interested in the following problem: do there exist infinitely many prime numbers $p$ such that $p^2|2^{n}+3$ for some natural number $n$? Some motivation: If we replace the function $2^n + 3$ with the $f(n)$ where $f \in \mathbb{Z}[x]$ is non-constant that this is true (follows Hensel lemma). So, it's rather natural to try proving this for…","stack exchange open-problems tag (mathoverflow, score 15, 2 answers)",,https://mathoverflow.net/questions/13444/prime-divisors-of-numbers-2n-3,
se-mo-135608,open,compute-checkable-small-cases,mathematics / number theory,"A conjecture by Euler about $8n+3$. Euler's conjecture: For any positive integer $n$, $8n+3$ can be represented as a sum $$8n+3=(2k-1)^2+2p,$$ where $k$ is a positive integer, and $p$ is a prime. I want to know whether there has been progress on the problem. Could you recommend some references? Thanks a","stack exchange open-problems tag (mathoverflow, score 16, 2 answers)",,https://mathoverflow.net/questions/135608/a-conjecture-by-euler-about-8n3,
se-mo-135765,open,needs-theory,mathematics / algebraic topology,"Open problems in algebraic topology and homotopy theory. Some time ago (I see it was initially written before 1999?) Mark Hovey assembled a list of open problems in algebraic topology. The list can be found here. Some of the problems I know about have been worked on quite a bit since the time of writing. The list is very good as is, but there must also be a few good additions since 1999. Can someone…","stack exchange open-problems tag (mathoverflow, score 32, 2 answers)",,https://mathoverflow.net/questions/135765/open-problems-in-algebraic-topology-and-homotopy-theory,
se-mo-135945,open,needs-theory,mathematics / set theory,"A question about the Axiom of Choice. Let AC denote the Axiom of Choice. Let PP denote the so-called ""Partition Principle"" which states that ""If S is a non-empty set and T is a non-empty set of pairwise disjoint subsets of S, then S can be mapped onto T"". It is well known that ""AC implies PP"" is provable in ZF, but the question of whether ""PP implies AC"" is provable in ZF has long…","stack exchange open-problems tag (mathoverflow, score 10, 1 answers)",,https://mathoverflow.net/questions/135945/a-question-about-the-axiom-of-choice,
se-mo-136157,open,needs-theory,mathematics / functional analysis,Set-theoretic reformulation of the invariant subspace problem. The invariant subspace problem (ISP) for Hilbert spaces asks whether every bounded linear operator $A$ on $l^2$ (with complex scalars) must have a closed invariant subspace other than $\{0\}$ and $l^2$. A subspace $E$ is invariant for $A$ if $A(E) \subseteq E$. Some time ago I noticed a reformulation that has a set-theoretic flavor. Let $P$ be the…,"stack exchange open-problems tag (mathoverflow, score 48, 0 answers)",,https://mathoverflow.net/questions/136157/set-theoretic-reformulation-of-the-invariant-subspace-problem,
se-mo-136351,open,needs-theory,mathematics / convexity,"Small quadrilaterals containing a given convex region. It is easy to prove that (*) Every convex planar set of area 1 is contained in a quadrilateral of area 2. It is also easy to see that statement (*) remains true if the constant 2 is replaced with a somewhat smaller one. Contest: Find such a constant, the smaller the better. Update: Reaching $\sqrt 2$ and even a strictly smaller value was proved by…","stack exchange open-problems tag (mathoverflow, score 8, 1 answers)",,https://mathoverflow.net/questions/136351/small-quadrilaterals-containing-a-given-convex-region,
se-mo-137287,open,needs-theory,mathematics / combinatorics,"Tiling the square with rectangles of small diagonals. For a given integer $k\ge3$, tile the unit square with $k$ rectangles so that the longest of the rectangles' diagonals be as short as possible. Call such a tiling optimal . The solutions are obvious in the easy cases when $k$ is the square of an integer and for a few small values of $k$ only (unpublished). In each of the solved cases, the sides of…","stack exchange open-problems tag (mathoverflow, score 16, 2 answers)",,https://mathoverflow.net/questions/137287/tiling-the-square-with-rectangles-of-small-diagonals,
se-mo-137884,open,needs-theory,mathematics / metric geometry,"Equiprojective polyhedra. Seeing Garabed Gulbenkian's question (which was inspired by Joel Hamkins' question ), reminds me of an analogous problem which I believe remains open, and which some might find intriguing. Define an equiprojective polyhedron $P \subset \mathbb{}R^3$ as one whose orthogonal projections—with the exception of projections in directions parallel to a…","stack exchange open-problems tag (mathoverflow, score 3, 1 answers)",,https://mathoverflow.net/questions/137884/equiprojective-polyhedra,
se-mo-137906,open,compute-checkable-small-cases,mathematics / functional analysis,"Do the banded operators check the invariant subspace problem? Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Invariant subspace problem : Let $T \in B(H)$. Is there a non-trivial closed $T$-invariant subspace? Remark : This problem is known for the Banach spaces in general, but still open for an Hilbert space. Let $(e_{n})_{n \in \mathbb{N}}$ be an…","stack exchange open-problems tag (mathoverflow, score 5, 0 answers)",,https://mathoverflow.net/questions/137906/do-the-banded-operators-check-the-invariant-subspace-problem,
se-mo-138368,open,needs-theory,mathematics / group theory,"Tarski Monster group with prime 5. Does the Tarski Monster group with prime 5 exist? I know that for 2 and 3, the group does not exist, but what about","stack exchange open-problems tag (mathoverflow, score 9, 0 answers)",,https://mathoverflow.net/questions/138368/tarski-monster-group-with-prime-5,
se-mo-140819,open,needs-theory,mathematics / graph theory,"Graphs with graphic imbalance sequences. Let $G$ be simple undirected graph and $e=uv\in E(G)$. The imbalance of the edge $e$ is the value $imb(e)=|d(u)-d(v)|$. Let $M_{G}$ denotes the imbalance sequence (or more correctly, multiset of all edge imbalances) of $G$. I can prove that if $T$ is a tree, then $M_{T}$ is graphic. However, in general case it isn't true. See A question on graphic…","stack exchange open-problems tag (mathoverflow, score 7, 0 answers)",,https://mathoverflow.net/questions/140819/graphs-with-graphic-imbalance-sequences,
se-mo-144080,open,compute-checkable-small-cases,mathematics / number theory,"Is $\varliminf_{n \rightarrow +\infty} |n \sin n| = 0$ correct, where $n$ is an integer? Is it true that $\varliminf_{n \rightarrow +\infty} |n \sin n| = 0$, where $n$ runs over the integers? The existence of the limes inferior follows from Dirichlet's approximation theorem , but the problem is to prove that it is","stack exchange open-problems tag (mathoverflow, score 13, 1 answers)",,https://mathoverflow.net/questions/144080/is-varliminf-n-rightarrow-infty-n-sin-n-0-correct-where-n-is-an,
se-mo-144081,open,compute-checkable-small-cases,mathematics / combinatorics,"Is the Steiner ratio Gilbert–Pollak conjecture still open? Gilbert-Pollak conjecture on the Steiner ratio: Consider a set $P$ of $n$ points on the euclidean plane. A shortest network interconnecting $P$ must be a tree, which is called a Steiner minimum tree and denoted by $SMT(P)$. A $SMT(P)$ may contain vertices not in $P$. Such vertices are called Steiner points , while vertices in $P$ are called…","stack exchange open-problems tag (mathoverflow, score 15, 2 answers)",,https://mathoverflow.net/questions/144081/is-the-steiner-ratio-gilbert-pollak-conjecture-still-open,
se-mo-145084,open,compute-checkable-small-cases,mathematics / number theory,"Euclidean real quadratic fields. It is known that, under GRH, a real quadratic field is Euclidean iff it is a UFD. So, assuming the conjecture of Gauss and GRH, we expect that there are infinitely many Euclidean real quadratic fields. I want to know if there are any approaches in showing that there are infinitely many Euclidean real quadratic fields by explicitly constructing the…","stack exchange open-problems tag (mathoverflow, score 12, 1 answers)",,https://mathoverflow.net/questions/145084/euclidean-real-quadratic-fields,
se-mo-145299,open,compute-checkable-small-cases,mathematics / logic,"Decidability of equality of elementary expressions. In the following definition the term expression is to be understood as a finite tree built from formal symbols without any predefined meaning assigned to them. Define the set $\mathcal{E}$ of elementary expressions as the minimal set such that: the expression $i$ is in $\mathcal{E}$ if $x\in\mathcal{E}$, then the expression…","stack exchange open-problems tag (mathoverflow, score 25, 1 answers)",,https://mathoverflow.net/questions/145299/decidability-of-equality-of-elementary-expressions,
se-mo-145467,open,needs-theory,mathematics / inequalities,"A challenging non homogenous fractional inequality. I have posted this question on Stackexchange but it has received no answer so far. It is a challenging generalization of several difficult inequalities, where none of the usual methods used in inequalities seems to work. I would like to know if someone has a deeper insight into what is happening here. Let $n\in\mathbb N$, $\lambda> 1$ and put…","stack exchange open-problems tag (mathoverflow, score 4, 0 answers)",,https://mathoverflow.net/questions/145467/a-challenging-non-homogenous-fractional-inequality,
se-mo-147407,open,compute-checkable-small-cases,mathematics / number theory,"A Question on 1, 2 ,3 Conjecture. The 1, 2, 3 conjecture is well-known: If $G$ is a simple graph which is not $K_2$ then one can assign a number among $1, 2, 3$ to every edge such that if we label each vertex with the sum of the numbers of edges incident with it then we obtain a proper vertex coloring. Question: Is the following weaker statement true? For any simple graph $G$…","stack exchange open-problems tag (mathoverflow, score 25, 1 answers)",,https://mathoverflow.net/questions/147407/a-question-on-1-2-3-conjecture,
se-mo-148648,open,needs-theory,mathematics / differential geometry,"Positively curved Riemannian manifolds. Let $M$ be a compact Riemannian manifold with positive sectional curvature whose universal covering space is diffeomorphic to $S^n$. Is $M$ diffeomorphic to a spherical space form? I know, by a theorem of Brendle and Schoen, if $M$ is a compact Riemannian manifold of dimension $n>3$ with pointwise $1/4$-pinched sectional curvature, then $M$ is…","stack exchange open-problems tag (mathoverflow, score 5, 1 answers)",,https://mathoverflow.net/questions/148648/positively-curved-riemannian-manifolds,
se-mo-149326,open,needs-theory,mathematics / banach spaces,"Second duals of Grothendieck spaces. The classical example of a Grothendieck space is $\ell_\infty$. It is also known that its even duals $\ell_\infty^{**}$, $\ell_\infty^{(4)}$, $\dots$ are Grothendieck spaces. (See, e.g ., this note for the proof of the Grothendieck property of these even duals, which uses the fact that these spaces are $C(K)$ spaces for large Stonian $K$-s.) Is it…","stack exchange open-problems tag (mathoverflow, score 12, 0 answers)",,https://mathoverflow.net/questions/149326/second-duals-of-grothendieck-spaces,
se-mo-15188,open,compute-checkable-small-cases,mathematics / metric geometry,"Is the ratio Perimeter/Area for a finite union of unit squares at most 4? Update: As I have just learned, this is called Keleti's perimeter area conjecture. Prove that if H is the union of a finite number of unit squares in the plane, then the ratio of the perimeter and the area of their union is at most four. Remarks. If the squares must be axis-aligned, then this is easy to prove. If we replace unit squares with unit…","stack exchange open-problems tag (mathoverflow, score 28, 3 answers)",,https://mathoverflow.net/questions/15188/is-the-ratio-perimeter-area-for-a-finite-union-of-unit-squares-at-most-4,
se-mo-153426,open,needs-theory,mathematics / number theory,"Floors of rationals to powers: Infinite number of primes? Let $r=a/b$ be a rational number in lowest terms, larger than $1$, and not an integer (so $b > 1$). Q . Does the sequence $$ \lfloor r \rfloor, \lfloor r^2 \rfloor, \lfloor r^3 \rfloor, \ldots, \lfloor r^n \rfloor, \ldots$$ always contain an infinite number of primes? For example, for $r=13/4$, $$(r,r^2,r^3,r^4,r^5, \ldots) = (3.25, 10.56, 34.33,…","stack exchange open-problems tag (mathoverflow, score 20, 2 answers)",,https://mathoverflow.net/questions/153426/floors-of-rationals-to-powers-infinite-number-of-primes,
se-mo-154025,open,compute-checkable-small-cases,mathematics / combinatorics,"A group-theoretic perspective on Frankl's union closed problem. Here is a group theoretic phrasing of a special case of the union closed conjecture: Question: Given a finite group $G$, is there an element of prime power order which is contained in at most half the subgroups of $G$? Motivation : Frankl's union closed sets conjecture has an equivalent phrasing in terms of lattices. It says that in every finite…","stack exchange open-problems tag (mathoverflow, score 37, 2 answers)",,https://mathoverflow.net/questions/154025/a-group-theoretic-perspective-on-frankls-union-closed-problem,
se-mo-1554,open,needs-theory,mathematics / differential geometry,"Point singularity of a Riemannian manifold with bounded curvature. Suppose you have an incomplete Riemannian manifold with bounded sectional curvature such that its completion as a metric space is the manifold plus one additional point. Does the Riemannian manifold structure extend across the point singularity? (Penny Smith and I wrote a paper on this many years ago, but we had to assume that no arbitrarily short…","stack exchange open-problems tag (mathoverflow, score 27, 5 answers)",,https://mathoverflow.net/questions/1554/point-singularity-of-a-riemannian-manifold-with-bounded-curvature,
se-mo-156076,open,compute-checkable-small-cases,mathematics / group theory,"An infinite, amenable, finitely presentable group with no non-trivial finite quotients. My question is a simple one: is there a group with the properties in the title? In the absence of the 'finitely presentable' hypothesis, an example is provided by Juschenko--Monod 's construction of a finitely generated, infinite, simple, amenable group. On the other hand, all the standard examples that I know of finitely presentable groups…","stack exchange open-problems tag (mathoverflow, score 15, 1 answers)",,https://mathoverflow.net/questions/156076/an-infinite-amenable-finitely-presentable-group-with-no-non-trivial-finite-quo,
se-mo-157184,open,needs-theory,mathematics / probability,"Convergence in $L^2$ of iterated expectations. Take a probability space $(\Omega,\mathcal{F},\mathbf{P})$ and random variable $X \in L^2(\Omega,\mathcal{F},\mathbf{P})$. Define the iterated expectations of X as follows: $X_0 = X$, and, inductively, $X_k = \mathbf{E}[X_{k-1} \, | \, \mathcal{G}_k]$, where $\mathcal{G}_k \subseteq \mathcal{F}$ is some $\sigma$-algebra. Assume that all the…","stack exchange open-problems tag (mathoverflow, score 9, 0 answers)",,https://mathoverflow.net/questions/157184/convergence-in-l2-of-iterated-expectations,
se-mo-158022,open,needs-theory,mathematics / classical analysis and odes,"An integral related to the Euler gamma function. The question is from the paper http://arxiv.org/abs/1312.7115 (A curious formula related to the Euler Gamma function, by Bakir Farhi): is it possible to express the integral $$\eta=2\int\limits_0^1 \ln{(\Gamma(x))}\cdot \sin{(2\pi x)}\,dx= 0.7687478924\ldots$$ in terms of the known mathematical constants as $\pi,\,e,\,\gamma,\;\ln{\pi},\,…","stack exchange open-problems tag (mathoverflow, score 11, 3 answers)",,https://mathoverflow.net/questions/158022/an-integral-related-to-the-euler-gamma-function,
se-mo-158039,open,needs-theory,mathematics / number theory,"Objections to and arguments for the simplicity of all Riemann zeros. It seems to be that the simplicity of all the zeros is quite widely accepted as a working hypotheses, and it is known that a positive proportion are as such. Titchmarsh explains in the last chapter on RH that the weak Mertens hypothesis, $M(x)=O(x^{1/2})$, implies that all zeros are simple. Though I think this is conjectured to be false by many…","stack exchange open-problems tag (mathoverflow, score 8, 3 answers)",,https://mathoverflow.net/questions/158039/objections-to-and-arguments-for-the-simplicity-of-all-riemann-zeros,
se-mo-159494,open,needs-theory,mathematics / graph theory,"Determine or estimate the number of maximal triangle-free graphs on $n$ vertices. Among the collections of the open problems of Paul Erdős on the website of Professor Fan Chung, there is one called ""number of triangle-free graphs"". http://www.math.ucsd.edu/~erdosproblems/erdos/newproblems/NumberOfTriangleFreeGraphs.html Open Problem: Determine or estimate the number of maximal triangle-free graphs on $n$ vertices. Is any one…","stack exchange open-problems tag (mathoverflow, score 18, 2 answers)",,https://mathoverflow.net/questions/159494/determine-or-estimate-the-number-of-maximal-triangle-free-graphs-on-n-vertices,
se-mo-161731,open,compute-checkable-small-cases,mathematics / logic,practical algorithms for np complete problems. Inspired by: Conjecture on NP-completeness of tesselation of Wang Tile up to finite size And the practicality of this topic (solving tessellation on a lattice): coloring in lattice Computational approach deciding whether a set of Wang Tile could tile the space up to some size Reference for Wang Tile I want to ask for practitioner what kind of…,"stack exchange open-problems tag (mathoverflow, score 5, 6 answers)",,https://mathoverflow.net/questions/161731/practical-algorithms-for-np-complete-problems,
se-mo-161947,open,compute-checkable-small-cases,mathematics / number theory,"What keeps asymptotic Goldbach's conjecture out of reach of current technology? Despite the rather recent progress in prime number theory (see the proof of the ternary Goldbach conjecture by H.A. Helfgott, and the striking result of Yitang Zhang improved by Tao, Maynard and others), to my knowledge, we're still unable to prove rigorously that there exists $N>0$ such that for all integer $n>N$, $2n$ is the sum of two primes.…","stack exchange open-problems tag (mathoverflow, score 10, 1 answers)",,https://mathoverflow.net/questions/161947/what-keeps-asymptotic-goldbachs-conjecture-out-of-reach-of-current-technology,
se-mo-162020,open,compute-checkable-small-cases,mathematics / combinatorics,"Which region in the plane with a given area has the most domino tilings? I just finished teaching a class in combinatorics in which I included a fairly easy upper bound on the number of domino tilings of a region in the plane as a function of its area. So this led to this question, which I suspect is an open problem: Does a $2n \times 2n$ square have the most domino tilings among all regions with area $4n^2$? I aired…","stack exchange open-problems tag (mathoverflow, score 59, 1 answers)",,https://mathoverflow.net/questions/162020/which-region-in-the-plane-with-a-given-area-has-the-most-domino-tilings,
se-mo-16214,open,compute-checkable-small-cases,mathematics / general,"Is there an associative metric on the non-negative reals? Recall that a function $f\colon X\times X \to \mathbb{R}_{\ge 0}$ is a metric if it satisfies: definiteness: $f(x,y) = 0$ iff $x=y$, symmetry: $f(x,y)=f(y,x)$, and the triangle inequality: $f(x,y) \le f(x,z) + f(z,y)$. A function $f\colon X\times X \to X$ is associative if it satisfies: associativity: $f(x,f(y,z)) = f(f(x,y),z)$. If…","stack exchange open-problems tag (mathoverflow, score 35, 2 answers)",,https://mathoverflow.net/questions/16214/is-there-an-associative-metric-on-the-non-negative-reals,
se-mo-162245,open,compute-checkable-small-cases,mathematics / combinatorics,"Conjecture of a subset of Wang tile which might be decidable. From the two papers proving the undecidability of Wang tile in 1966 by Berger and in 1971 by RM Robinson, the tiles used in proving undecidability has a general common feature: The left color and right color do not uniquely determine the top color and bottom color. The tiles involved in practice in material science and physics for one big problem…","stack exchange open-problems tag (mathoverflow, score 3, 2 answers)",,https://mathoverflow.net/questions/162245/conjecture-of-a-subset-of-wang-tile-which-might-be-decidable,
se-mo-163301,open,needs-theory,mathematics / mathematical physics,"What is Chern-Simons theory expected to assign to a point? Let $G$ be a compact, connected, (simply connected?) Lie group and let $k \in H^4(BG, \mathbb{Z})$ be a cohomology class. Witten showed, at a physical level of rigor, that this data determines a $3$-dimensional topological quantum field theory (going down to surfaces), Chern-Simons theory. Later other authors (Reshetikhin-Turaev, ???) described…","stack exchange open-problems tag (mathoverflow, score 22, 3 answers)",,https://mathoverflow.net/questions/163301/what-is-chern-simons-theory-expected-to-assign-to-a-point,
se-mo-16332,open,compute-checkable-small-cases,mathematics / algebraic geometry,"MNOP conjecture. Let $X$ be a smooth, projective, Calabi-Yau 3-fold (CY makes the exposition more elegant, I don't think it is necessary). To define Gromov-Witten invariants, we consider moduli spaces of stable maps , $\bar{M}_g(X, \beta),$ indexed by the genus $g$ of the source curve and curve class $\beta$ of its image in $X$ . These moduli spaces admit virtual…","stack exchange open-problems tag (mathoverflow, score 17, 0 answers)",,https://mathoverflow.net/questions/16332/mnop-conjecture,
se-mo-16335,open,needs-theory,mathematics / smooth manifolds,"A generalization of Cauchy's mean value theorem. The following simple theorem is known as Cauchy's mean value theorem. Let $\gamma$ be an immersion of the segment $[0,1]$ into the plane such that $\gamma(0) \ne \gamma(1)$. Then there exists a point such that the tangent line at that point is parallel to the line passing through $\gamma(0)$ and $\gamma(1)$. So the boundary values of an immersion…","stack exchange open-problems tag (mathoverflow, score 11, 2 answers)",,https://mathoverflow.net/questions/16335/a-generalization-of-cauchys-mean-value-theorem,
se-mo-16341,open,needs-theory,mathematics / number theory,"On Polynomials dividing Exponentials. EDIT, May 2015: in the second edition of the relevant book , the question was corrected, a single number had been mis-typed. The corrected question (thanks to Max) is to find all positive integer pairs such that $$ m^2 - 1 \; | \; 3^m + \left( n! - 2 \right)^m. $$ $$ $$ The version that caused me all that misery was $ m^2 - 1 \; | \; 3^m + \left(…","stack exchange open-problems tag (mathoverflow, score 14, 6 answers)",,https://mathoverflow.net/questions/16341/on-polynomials-dividing-exponentials,
se-mo-163845,open,needs-theory,mathematics / graph theory,"Integral straight-line embeddings of planar graphs. Wikipedia says (in the article on Fáry's theorem) , ""Heiko Harborth raised the question of whether every planar graph has a straight line representation in which all edge lengths are integers. The answer remains unknown as of 2009."" The reference is this: Kemnitz, Arnfried, and Heiko Harborth. ""Plane integral drawings of planar graphs."" Discrete…","stack exchange open-problems tag (mathoverflow, score 7, 1 answers)",,https://mathoverflow.net/questions/163845/integral-straight-line-embeddings-of-planar-graphs,
se-mo-16410,open,needs-theory,mathematics / geometric topology,"$\operatorname{SL}_2(\mathbb R)$ Casson invariant? $\DeclareMathOperator\SL{SL}$ Casson's invariant is an invariant of a homology 3-sphere, obtained by “counting” representations of the fundamental group into $\operatorname{SU}(2)$ . I was wondering if there is an analogous invariant counting representations into $\SL(2,\mathbb R)$ ? Curtis has an invariant counting representations into…","stack exchange open-problems tag (mathoverflow, score 16, 2 answers)",,https://mathoverflow.net/questions/16410/operatornamesl-2-mathbb-r-casson-invariant,
se-mo-164129,open,needs-theory,mathematics / number theory,"improving known bounds for Pierce expansions; cash prize. Here's a problem that I thought of back in 1978 or so, and only a little progress has been made on it since then. I still think about it from time to time, but probably not that many people have thought about it. As I'm getting older, I now offer a cash prize for anyone who can make new progress. The problem concerns the number of steps in an…","stack exchange open-problems tag (mathoverflow, score 48, 1 answers)",,https://mathoverflow.net/questions/164129/improving-known-bounds-for-pierce-expansions-cash-prize,
se-mo-164665,open,compute-checkable-small-cases,mathematics / group theory,"Is there a precise notion of ""almost all"" such that almost all finite groups are Galois groups of extensions of the rationals? I vainly tried to define a notion of ""almost solvable group"" such that every ""almost solvable group"" is the Galois group of a finite extension of the rationals, but I can't figure out the right way to do so, so I want to ask a probably less ambitious question. Is there a precise notion of ""almost all"" such that almost all finite groups are Galois…","stack exchange open-problems tag (mathoverflow, score 6, 1 answers)",,https://mathoverflow.net/questions/164665/is-there-a-precise-notion-of-almost-all-such-that-almost-all-finite-groups-are,
se-mo-164983,open,compute-checkable-small-cases,mathematics / combinatorics,"Generating random finite groups. I would like a method to efficiently generate a random finite group of a given order $n$ . If there are $g(n)$ non-isomorphic groups of order $n$ , ideally each group would occur with probability $1/g(n)$ . So if $n=64$ , each of the $267$ groups would be generated with equal probability. ( $g(n)$ is A000001 in OEIS.) Groups of order $n=2^k$ would…","stack exchange open-problems tag (mathoverflow, score 21, 2 answers)",,https://mathoverflow.net/questions/164983/generating-random-finite-groups,
se-mo-16818,open,compute-checkable-small-cases,mathematics / differential geometry,"Are there Ricci-flat riemannian manifolds with generic holonomy? This may well be an open problem, I'm not sure. In Berger's classification (refined by Simons, Alekseevsky, Bryant,...) of the holonomy representations of irreducible non-symmetric complete simply-connected riemannian manifolds, there are some cases which imply Ricci-flatness: namely, $\mathrm{SU}(n)$ (Calabi-Yau) in dimension $2n$,…","stack exchange open-problems tag (mathoverflow, score 29, 2 answers)",,https://mathoverflow.net/questions/16818/are-there-ricci-flat-riemannian-manifolds-with-generic-holonomy,
se-mo-17058,open,needs-theory,mathematics / combinatorics,"Factorials in Pascal's triangle. I asked this question of Keith Conrad, and he suggested that I try posting here. One of my students observed that the only instances of factorials in the interior of Pascal's triangle are $\binom{4}{2}=3!$ and $\binom{10}{3}=\binom{16}{2}=5!$ . I checked the first $500$ rows, and he's right up to that point. This is a special case of the…","stack exchange open-problems tag (mathoverflow, score 39, 5 answers)",,https://mathoverflow.net/questions/17058/factorials-in-pascals-triangle,
se-mo-171803,open,needs-theory,mathematics / metric geometry,"Packing space by cones: Translates best? Let $C$ be a right circular cone, the convex hull of a unit-radius disk and a point directly above the disk center at height $h$. Is the most dense packing of $\mathbb{R}^3$ achieved by translates of $C$ and its negative, * $-C$, i.e., upside-down $C$? Or is it known that rotations of the cones permit achieving a denser packing? * Barany, Imre,…","stack exchange open-problems tag (mathoverflow, score 4, 0 answers)",,https://mathoverflow.net/questions/171803/packing-space-by-cones-translates-best,
se-mo-172388,open,compute-checkable-small-cases,mathematics / algebraic geometry,"A special case of the integer Hodge conjecture. Let $X$ be a projective complex manifold of dimension $n$. Are torsion cohomology classes in $H^{2n-2}(X,\mathbb{Z})$ algebraic? (We may assume, without loss of generality, that $n=3$, because of the Lefschetz hyperplane theorem.) I know that torsion classes (of even codimension) aren't always algebraic; the first counterexamples were found by…","stack exchange open-problems tag (mathoverflow, score 11, 1 answers)",,https://mathoverflow.net/questions/172388/a-special-case-of-the-integer-hodge-conjecture,
se-mo-17257,open,compute-checkable-small-cases,mathematics / mathematical physics,"What's the current state of Yang–Mills mass gap question? What's the current state of Yang–Mills mass gap question, is there any place that does this problem? Especially I want to know if there is any progress (out of that mentioned in the introduction article by Witten and Jaffe). Is it too hard for a mathematician?","stack exchange open-problems tag (mathoverflow, score 11, 3 answers)",,https://mathoverflow.net/questions/17257/whats-the-current-state-of-yang-mills-mass-gap-question,
se-mo-17258,open,compute-checkable-small-cases,mathematics / matrices,"What's known about the 3rd coefficient in the BMV conjecture? The most appealing statement of the Bessis-Moussa-Villani conjecture is as follows: Conjecture: For all Hermitian positive semidefinite $n\times n$ matrices $A$ and $B$, and all positive integer $m$, the polynomial function $$t \in \mathbb{R}\mapsto g(t) \equiv tr[(A + t B)^m] = \sum\limits_{ k=0}^m a_kt^k$$ has only nonnegative coefficients $a_k,…","stack exchange open-problems tag (mathoverflow, score 3, 2 answers)",,https://mathoverflow.net/questions/17258/whats-known-about-the-3rd-coefficient-in-the-bmv-conjecture,
se-mo-172626,open,compute-checkable-small-cases,mathematics / algebraic topology,"Is there a non-zero ghost map between finite suspension spectra? A morphism $f\colon X\to Y$ of spectra such that for every integer $n$ the induced map $\pi_n(f)\colon\pi_n(X)\to\pi_n(Y)$ on stable homotopy groups is zero is called a ghost map . Not every ghost map $f$ is the zero object of the abelian group $\operatorname{Hom}_{SH}(X,Y)$ (more precisely, its image $\bar f\in \operatorname{Hom}_{SH}(X,Y)$ in…","stack exchange open-problems tag (mathoverflow, score 14, 1 answers)",,https://mathoverflow.net/questions/172626/is-there-a-non-zero-ghost-map-between-finite-suspension-spectra,
se-mo-17313,open,compute-checkable-small-cases,mathematics / discrete geometry,"Is it possible to dissect a disk into congruent pieces, so that a neighborhood of the origin is contained within a single piece? Problem: is it possible to dissect the interior of a circle into a finite number of congruent pieces (mirror images are fine) such that some neighbourhood of the origin is contained in just one of the pieces? It may be conceivable that there is some dissection into immeasurable sets that does this. So a possible additional constraint would be that…","stack exchange open-problems tag (mathoverflow, score 31, 2 answers)",,https://mathoverflow.net/questions/17313/is-it-possible-to-dissect-a-disk-into-congruent-pieces-so-that-a-neighborhood-o,
se-mo-173446,open,compute-checkable-small-cases,mathematics / banach spaces,"Subspaces of $l_{1}$ are not Lipschitz complemented in $l_{1}$. I have thought about the following question for several years. This question may be stupid or not interesting. My question is: Is there a subspace $U$ of $l{_1}$ such that the quotient $l_{1}/U$ is isomorphic to $L_{1}(\mu)$ for some measure $\mu$, but $U$ is not Lipschitz complemented in","stack exchange open-problems tag (mathoverflow, score 10, 1 answers)",,https://mathoverflow.net/questions/173446/subspaces-of-l-1-are-not-lipschitz-complemented-in-l-1,
se-mo-17532,open,needs-theory,mathematics / category theory,"Does linearization of categories reflect isomorphism? Given a category $C$ and a commutative ring $R$, denote by $RC$ the $R$-linearization: this is the category enriched over $R$-modules which has the same objects as $C$, but the morphism module between two objects $x$ and $y$ is the free $R$-module on $\operatorname{Hom}_C(x,y)$. Thus in $RC$ we allow arbitrary $R$-linear combinations of morphisms…","stack exchange open-problems tag (mathoverflow, score 66, 3 answers)",,https://mathoverflow.net/questions/17532/does-linearization-of-categories-reflect-isomorphism,
se-mo-17560,open,compute-checkable-small-cases,mathematics / number theory,"If $2^x $and $3^x$ are integers, must $x$ be as well? I'm fascinated by this open problem (if it is indeed still that) and every few years I try to check up on its status. Some background: Let $x$ be a positive real number. If $n^x$ is an integer for every $n \in \mathbb{N}$ then $x$ must be an integer. This is a fun little puzzle. If $2^x$, $3^x$ and $5^x$ are integers then $x$ must be an integer.…","stack exchange open-problems tag (mathoverflow, score 161, 5 answers)",,https://mathoverflow.net/questions/17560/if-2x-and-3x-are-integers-must-x-be-as-well,
se-mo-176923,open,needs-theory,mathematics / representation theory,"Update on list of open problems for Cherednik/Symplectic Reflection Algebras. Background: There are two lists of open problems about Cherednik or Symplectic Reflection Algebras from 2007: Ian Gordon's Problems, Chapter 9 in Symplectic Reflection Algebras , and Ginzburg & Etingof's list given at a workshop in 2007 here . Gordon titled these ""Problems for next year"". Now, more than years later, some of them are certainly…","stack exchange open-problems tag (mathoverflow, score 11, 2 answers)",,https://mathoverflow.net/questions/176923/update-on-list-of-open-problems-for-cherednik-symplectic-reflection-algebras,
se-mo-179206,open,compute-checkable-small-cases,mathematics / order theory,"reduction to np hard ordering problem. I am trying to show a reduction from a problem of ordering problem to an np-hard problem that has approximation poly-time algorithm. My problem is: I have M auctions and in each auction I have N bidders. A bidder can bid in all the auctions until he wins one, but if he won one of the auctions he can't win other items. assuming that the auctioneer…","stack exchange open-problems tag (mathoverflow, score 3, 1 answers)",,https://mathoverflow.net/questions/179206/reduction-to-np-hard-ordering-problem,
se-mo-179790,open,compute-checkable-small-cases,mathematics / set theory,"Is the fixed point property for posets preserved by products? Recall that a partially ordered set (poset) $P$ has the fixed point property (FPP) if any order preserving function $f:P\longrightarrow P$ has a fixed point. Theorem. Suppose $P$ and $Q$ are posets with the FPP and at least one of them is finite. Then $(P\times Q)$ has FPP. Note: $(a,b)\le(c,d)$ if and only if $a\le c$ and $b\le d$. Question.…","stack exchange open-problems tag (mathoverflow, score 31, 3 answers)",,https://mathoverflow.net/questions/179790/is-the-fixed-point-property-for-posets-preserved-by-products,
se-mo-179898,open,needs-theory,mathematics / differential geometry,"Does $S^4$ have a ""symplecto-homeomorphic"" structure? The 4-sphere cannot be a symplectic manifold. In particular, it does not admit an atlas whose transition maps are symplectomorphisms $(\mathbb{R}^4,\omega_\text{std})\to(\mathbb{R}^4,\omega_\text{std})$. Eliashberg-Gromov proved that if the $C^0$-limit of symplectomorphisms is a diffeomorphism, then it is a symplectomorphism. We can now ask about…","stack exchange open-problems tag (mathoverflow, score 17, 0 answers)",,https://mathoverflow.net/questions/179898/does-s4-have-a-symplecto-homeomorphic-structure,
se-mo-182,open,compute-checkable-small-cases,mathematics / subfactors,"Is there a finite-index finite-depth II$_1$ subfactor which is more than $7$-super-transitive? Background: See Noah and Emily's posts on subfactors and planar algebras on the Secret Blogging Seminar. There are plenty of examples of $3$-super-transitive (3-ST) subfactors; Haagerup, $S_4 < S_5$, and others. There's exactly one known example of a $5$-ST subfactor, the Haagerup-Asaeda subfactor, and one $7$-ST subfactor, the extended Haagerup…","stack exchange open-problems tag (mathoverflow, score 8, 0 answers)",,https://mathoverflow.net/questions/182/is-there-a-finite-index-finite-depth-ii-1-subfactor-which-is-more-than-7-sup,
se-mo-18264,open,compute-checkable-small-cases,mathematics / functional analysis,"$\ell^2$, $\ell^3$, and $\ell^4$: a possible fixed point result? The question below is related to the classical Browder-Goehde-Kirk fixed point theorem . Let $K$ be the closed unit ball of $\ell^{2}$ , and let $T:K\rightarrow K$ be a mapping such that $$\Vert Tx-Ty\Vert _{\ell^{4}}\leq\Vert x-y\Vert _{\ell^{3}}$$ for all $x,y\in K$ . Is it true that $T$ has fixed","stack exchange open-problems tag (mathoverflow, score 86, 2 answers)",,https://mathoverflow.net/questions/18264/ell2-ell3-and-ell4-a-possible-fixed-point-result,
se-mo-186534,open,needs-theory,mathematics / probability,"Randomly placing nonoverlapping unit cuboids. Suppose one places unit cuboids of dimension $d$ with min-corners uniformly distributed to lie in $[0,n]^d$, but with cuboid (strict) overlap forbidden. At some point, the region is ""saturated,"" without room for more cuboids to fit. For example, for $d=2$ squares in $[0,10]^2$, it appears that about $63$ squares on average will fit (whereas…","stack exchange open-problems tag (mathoverflow, score 9, 0 answers)",,https://mathoverflow.net/questions/186534/randomly-placing-nonoverlapping-unit-cuboids,
se-mo-187421,open,needs-theory,mathematics / differential geometry,"Inverted pair of complex analytic families. I read the following ""problem"" in an old set of notes of Morrow and Kodaira which focused on deformations of complex manifolds: Find a pair of complex analytic families $\lbrace M_t\rbrace$ and $\lbrace N_t\rbrace$ with $|t|<1$ such that $M_t=N_0$ for $t\ne 0$ and $N_t=M_0$ for $t\ne0$, with $M_0\ne N_0$. (Every manifold assumed compact.) As of…","stack exchange open-problems tag (mathoverflow, score 6, 1 answers)",,https://mathoverflow.net/questions/187421/inverted-pair-of-complex-analytic-families,
se-mo-187680,open,needs-theory,mathematics / combinatorics,"Reduction argument from a general vertex set V(G) to a prime power in Prof. Keevash's proof on the Existence of Designs. The proof flow of the paper ""On the Existence of Designs"" by Prof. Keevash as I understand it is the following: -- Reduction from the general case to $V = \mathbb{F}_{p^a}$ (Lemma 6.3) -- Covering all the linearly dependent r-sets of $V = \mathbb{F}_{p^a}$ using $\Phi'$. This also covers ""bounded"" number of r-sets which have linearly independent…","stack exchange open-problems tag (mathoverflow, score 4, 0 answers)",,https://mathoverflow.net/questions/187680/reduction-argument-from-a-general-vertex-set-vg-to-a-prime-power-in-prof-keev,
se-mo-191191,open,needs-theory,mathematics / combinatorics,Open problems in compressed sensing. What are the main open problems in compressed sensing? I am interested in theoretical as well as in numerical point of,"stack exchange open-problems tag (mathoverflow, score 3, 1 answers)",,https://mathoverflow.net/questions/191191/open-problems-in-compressed-sensing,
se-mo-191386,open,needs-theory,mathematics / number theory,"Ruth-Aaron triples, etc. A Ruth-Aaron pair is two numbers $(n,n+1)$ such that their sum of prime factors is equal, counting repeated prime factors. (The name refers to Hank Aaron's 715 homeruns surpassing Babe Ruth's 714!) So $$15 = 3 \cdot 5 \;;\; 3+5 = 8$$ $$16 = 2^4 \;;\; 2+2+2+2 = 2 \cdot 4 = 8$$ The notion can be generalized: $$417,162 = 2 \cdot 3 \cdot 251 \cdot 277…","stack exchange open-problems tag (mathoverflow, score 5, 1 answers)",,https://mathoverflow.net/questions/191386/ruth-aaron-triples-etc,
se-mo-196033,open,compute-checkable-small-cases,mathematics / group theory,Can any finite lattice be realized as an intermediate subgroups lattice? Let $G$ be a finite group and $H$ a subgroup. Let $\mathcal{L}(H \subset G )$ be the lattice of all the intermediate subgroups between $H$ and $G$ . Question: Can any finite lattice be realized as an intermediate subgroups lattice? Remark : It's true for all the finite distributive lattices (see theorem 2.1 here,"stack exchange open-problems tag (mathoverflow, score 10, 2 answers)",,https://mathoverflow.net/questions/196033/can-any-finite-lattice-be-realized-as-an-intermediate-subgroups-lattice,
se-mo-1973,open,compute-checkable-small-cases,mathematics / differential geometry,"Is there a complex structure on the 6-sphere? I don't know who first asked this question, but it's a question that I think many differential and complex geometers have tried to answer because it sounds so simple and fundamental. There are even a number of published proofs that are not taken seriously, even though nobody seems to know exactly why they are","stack exchange open-problems tag (mathoverflow, score 109, 12 answers)",,https://mathoverflow.net/questions/1973/is-there-a-complex-structure-on-the-6-sphere,
se-mo-1975,open,needs-theory,mathematics / differential geometry,Is a smooth closed surface in Euclidean 3-space rigid? Classical theorem of Cohn-Vossen: A closed convex surface in Euclidean 3-space cannot be deformed isometrically. Robert Connelly found an example of a polyhedral surface that can be deformed isometrically. A metal hinged model of it can be found at IHES. But what about an arbitrary not-necessarily-convex smooth closed surface? Is it necessarily…,"stack exchange open-problems tag (mathoverflow, score 15, 2 answers)",,https://mathoverflow.net/questions/1975/is-a-smooth-closed-surface-in-euclidean-3-space-rigid,
se-mo-201419,open,needs-theory,mathematics / number theory,"What is the status on questions related to Bhargava's factorial function? In Manjul Bhargava's The Factorial Function and Generalizations he motivates a new type of factorial $n!_S$ using by generalizing a few theorems like: For $k, l \in \mathbb{Z}$, we have $k! \times l!$ divides $(k+l)!$. For any primitive polynomial $f(x) \in \mathbb{Z}[x]$ with $\deg f = k$ then $\mathrm{gcd}\{ f(a): a \in \mathbb{Z}\}$ divides…","stack exchange open-problems tag (mathoverflow, score 4, 0 answers)",,https://mathoverflow.net/questions/201419/what-is-the-status-on-questions-related-to-bhargavas-factorial-function,
se-mo-201922,open,compute-checkable-small-cases,mathematics / dynamical systems,"*The* open problem in General Relativity? Q . Is there a single, clear mathematical question that has emerged as the open problem in General Relativity? I ask this on the ~100 th anniversary of Einstein's (4-page!) 1915 paper, ""Die Feldgleichungen der Gravitation,"" Preussische Akademie der Wissenschaften , Sitzungsberichte, 1915 (part 2), 844–847. ( Wikisource:The Field Equations of…","stack exchange open-problems tag (mathoverflow, score 19, 2 answers)",,https://mathoverflow.net/questions/201922/the-open-problem-in-general-relativity,
se-mo-20200,open,compute-checkable-small-cases,mathematics / random matrices,"distribution of degree of minimum polynomial for eigenvalues of random matrix with elements in finite field. This is an attempt to extend the current full fledged random matrix theory to fields of positive characteristics. So here is a possible setup for the problem: Let $A_{n,p}$ be an $n \times n$ matrix with entries iid taking values uniformly in $F_p$. Then one should be able to find its eigenvalues together with multiplicities, which might lie in…","stack exchange open-problems tag (mathoverflow, score 7, 1 answers)",,https://mathoverflow.net/questions/20200/distribution-of-degree-of-minimum-polynomial-for-eigenvalues-of-random-matrix-wi,
se-mo-204361,open,needs-theory,mathematics / functional analysis,"Open problems in Banach spaces, universality. I have gathered a list of universality problems in Banach spaces which have been solved: 1.The non existence of a separable reflexive space universal for the class of separable reflexive spaces. 2.If a space is universal for the class of separable reflexive spaces, then it is universal for the class of separable Banach spaces. 3.There is a…","stack exchange open-problems tag (mathoverflow, score 6, 1 answers)",,https://mathoverflow.net/questions/204361/open-problems-in-banach-spaces-universality,
se-mo-20497,open,needs-theory,mathematics / algebraic geometry,"Is it known that the ring of periods is not a field? I have just learned here that we know numbers that are not periods; is it known meanwhile that the ring of periods is not a field? I know that it is conjectured that $1/\pi$ is not a period, but the existence of a period whose inverse is not a period seems to be still open. Is this correct? More generally: is it believed that the unit group of the…","stack exchange open-problems tag (mathoverflow, score 76, 2 answers)",,https://mathoverflow.net/questions/20497/is-it-known-that-the-ring-of-periods-is-not-a-field,
se-mo-209190,open,needs-theory,mathematics / hyperplane arrangements,"Open problems in hyperplane/subspace arrangements? What are some open problems in hyperplane/subspace arrangements, preferably of the combinatorial algebraic topology kind, and where can one read about them? That is, where are they discussed, and where can one learn the necessary background knowledge to understand them properly and in context? I have been working one one such problem for a while,…","stack exchange open-problems tag (mathoverflow, score 10, 2 answers)",,https://mathoverflow.net/questions/209190/open-problems-in-hyperplane-subspace-arrangements,
se-mo-21003,open,compute-checkable-small-cases,mathematics / number theory,"Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$? Is there any polynomial $f(x,y)\in{\mathbb Q}[x,y]{}$ such that $f\colon\mathbb{Q}\times\mathbb{Q} \rightarrow\mathbb{Q}$ is a","stack exchange open-problems tag (mathoverflow, score 547, 3 answers)",,https://mathoverflow.net/questions/21003/polynomial-bijection-from-mathbb-q-times-mathbb-q-to-mathbb-q,
se-mo-212764,open,compute-checkable-small-cases,mathematics / plane geometry,"Why is it so hard to prove Toeplitz' conjecture? I'm a layman in mathematics, so please excuse me in advance for anything in this question that may be inappropriate :D. Well: Four years ago, I was reading (and working to solve the puzzles on) Winkler's Mathematical Puzzles: A Connoisseur's Collection , and in the section ""Unsolved Puzzles"", there was the problem ""Squaring the Lake"" at p. 143:…","stack exchange open-problems tag (mathoverflow, score 25, 1 answers)",,https://mathoverflow.net/questions/212764/why-is-it-so-hard-to-prove-toeplitz-conjecture,
se-mo-217567,open,needs-theory,mathematics / classical analysis and odes,"Functions on [0,1] with positive fractional series coefficients and symmetric under x->1-x. Suppose a function $g(x)$ has a convergent power series expansion with real (not necessarily integer or rational) exponents on $[0,1]$ of the form $$ g(x)=\frac{1}{x^\delta}(1+\sum_{i=1}^\infty c_i x^{\Delta_i}) $$ where $\delta>0$, $0<\Delta_1<\Delta_2<\ldots$ and, crucially, $c_i>0$. Suppose further that $$ g(x)=g(1-x) $$ What is known about…","stack exchange open-problems tag (mathoverflow, score 6, 0 answers)",,https://mathoverflow.net/questions/217567/functions-on-0-1-with-positive-fractional-series-coefficients-and-symmetric-un,
se-mo-218508,open,compute-checkable-small-cases,mathematics / set theory,"Sets not containing the vertices of unit triangles (Question posed by Erdős). Following this post , I have been thinking about the problem posed by Erdős, Does there exist a constant $c > 0$ such that every subset $A$ of the plane of area more than $c$ contains the vertices of a triangle of unit area? I think the spirit of the problem justifies assuming $A$ is measurable (counterexamples exist for non-measurable sets, see…","stack exchange open-problems tag (mathoverflow, score 6, 1 answers)",,https://mathoverflow.net/questions/218508/sets-not-containing-the-vertices-of-unit-triangles-question-posed-by-erd%c5%91s,
se-mo-218884,open,needs-theory,mathematics / metric geometry,"Bang's open question strengthening Tarski's planks problem. Tarski's Planks problem , solved by Thøger Bang in 1951, says (in a simplified $\mathbb{R}^2$ version) that it requires ""planks"" (parallel strips) of total width $\ge d$ in order to completely cover a disk of diameter $d$. The last paragraph of Bang's short paper poses an open problem: (Snapshot from Bang's paper.) Does anyone know if this problem…","stack exchange open-problems tag (mathoverflow, score 10, 0 answers)",,https://mathoverflow.net/questions/218884/bangs-open-question-strengthening-tarskis-planks-problem,
se-mo-219344,open,compute-checkable-small-cases,mathematics / algebraic topology,"Gerstenhaber conjecture for free loop space. I- Is the following statement still a conjecture see this article ? Conjecture (?) Let $M$ be a simply connected compact oriented $d$-manifold (smooth), then $HH^{\ast}(C^{\ast}(M))$ the Hochschild cohomology of cochain complexes associated to $M$ is isomorphic as a Gerstenhaber algebra to $H_{\ast+d}(LM)$ the $d$-shifted homology of the free loop…","stack exchange open-problems tag (mathoverflow, score 12, 2 answers)",,https://mathoverflow.net/questions/219344/gerstenhaber-conjecture-for-free-loop-space,
se-mo-219552,open,needs-theory,mathematics / set theory,"A variant of Freiling's Axiom of Symmetry and a weak form of the Continuum Hypothesis in models where all sets of reals are Lebesgue measurable. Consider the following variant of Freiling's Axiom of Symmetry, $\mathsf{AS}$ , which will be denoted $A_{< 2^{\aleph_0}}$: given any function $f$ from $\mathbb{R}$ into the families of of subsets of $\mathbb{R}$ of size $< 2^{\aleph_0}$ there are $x_1,x_2 \in \mathbb{R}$ such that $x_1 \notin f(x_2)$ and $x_2 \notin f(x_1)$. Does $\mathsf{ZF} +…","stack exchange open-problems tag (mathoverflow, score 5, 2 answers)",,https://mathoverflow.net/questions/219552/a-variant-of-freilings-axiom-of-symmetry-and-a-weak-form-of-the-continuum-hypot,
se-mo-22,open,compute-checkable-small-cases,mathematics / number theory,"Can $N^2$ have only digits 0 and 1, other than $N=10^k$? Pablo Solis asked this at a recent 20 questions seminar at Berkeley. Is there a positive integer $N$, not of the form $10^k$, such that the digits of $N^2$ are all 0's and 1's? It seems very unlikely, but I don't have a proof. It's easy to see that such a number must end in 1 or 9, and then easy to see that it must end in 01, 49, 51 or 99, and you…","stack exchange open-problems tag (mathoverflow, score 53, 5 answers)",,https://mathoverflow.net/questions/22/can-n2-have-only-digits-0-and-1-other-than-n-10k,
se-mo-224711,open,needs-theory,mathematics / differential geometry,"All compact surfaces $S\subseteq \mathbb{R}^3$ are rigid? Recently I've come across a lecture in differential geometry by Fernando Codá (in portuguese!) in which he stated that the following problem is (at least, up to 2014) open: Given $S\subseteq\mathbb{R}^3$ a smooth (i.e. $C^\infty$) compact boundaryless connected surface (therefore orientable), let $f:S\times[0,1]\to \mathbb{R}^3$ be a…","stack exchange open-problems tag (mathoverflow, score 20, 3 answers)",,https://mathoverflow.net/questions/224711/all-compact-surfaces-s-subseteq-mathbbr3-are-rigid,
se-mo-225572,open,compute-checkable-small-cases,mathematics / number theory,"On cubic reciprocity for $x^3+y^3+z^3 = 996$? I. The Diophantine equation, $$x^3+y^3+z^3 = 3w^3\tag1$$ with $x\geq y \geq z$ and $w=1$ has only two known solutions, namely $1,1,1$ and $4,4,-5$. Are there larger ones? As Noam Elkies points out in this post , Cassels in a 1985 paper showed that $w=1$ must have, $$x\equiv y\equiv z \bmod 9\tag2$$ For general $w$, Heath-Brown in a 1992 paper…","stack exchange open-problems tag (mathoverflow, score 14, 1 answers)",,https://mathoverflow.net/questions/225572/on-cubic-reciprocity-for-x3y3z3-996,
se-mo-226001,open,compute-checkable-small-cases,mathematics / number theory,"Some Questions on the Collatz conjecture (reexpressed as ""equivalence relation""). The set of all positive whole numbers is denoted by $\mathbb{N}_+$. Let $f\colon\ \mathbb{N}_+\to\mathbb{N}_+:n\mapsto \begin{cases}\frac{n}{2}&\text{$n$ even}\\3n+1&\text{$n$ odd}\end{cases}$. Conjecture (Collatz). $\forall n\in\mathbb{N}_+.\ \exists N\in\mathbb{N}.\ f^N(n)=1$. Let $m,n\in\mathbb{N}_+$. We define: $m\sim n:\iff\exists N_1,…","stack exchange open-problems tag (mathoverflow, score 11, 1 answers)",,https://mathoverflow.net/questions/226001/some-questions-on-the-collatz-conjecture-reexpressed-as-equivalence-relation,
se-mo-226580,open,compute-checkable-small-cases,mathematics / logic,Does every model of $I\Delta_0$ has an end extension to a model of $I\Delta_0+\Omega_1$? Does every model of $I\Delta_0$ has an end extension to a model of $I\Delta_0+\Omega_1$? In End extensions of models of linearly bounded arithmetic paper the author said this problem is open. I want know is there any progress on this problem?,"stack exchange open-problems tag (mathoverflow, score 8, 0 answers)",,https://mathoverflow.net/questions/226580/does-every-model-of-i-delta-0-has-an-end-extension-to-a-model-of-i-delta-0-o,
se-mo-227958,open,definition-dispute,mathematics / number theory,"Why is Kronecker's Jugendtraum only for abelian extensions? Why is Kronecker's Jugendtraum only for abelian and not for more general extensions of number fields? Wikipedia, Hilbert's Twelfth Problem","stack exchange open-problems tag (mathoverflow, score 5, 1 answers)",,https://mathoverflow.net/questions/227958/why-is-kroneckers-jugendtraum-only-for-abelian-extensions,
se-mo-230174,open,needs-theory,mathematics / number theory,"If $N = qn^2$ is an odd perfect number with $\gcd(q,n)=1$, is it possible to have $q + 1 = \sigma(n)$? The title says it all. Question If $N = qn^2$ is an odd perfect number with Euler prime $q$ and $\gcd(q,n)=1$ , is it possible to have $q + 1 = \sigma(n)$ ? Heuristic From the Descartes spoof, with quasi-Euler prime $q_1$ : $$n_1 = 3003 < \sigma(n_1) = 5376 < q_1 = 22021$$ So it appears that it might be possible to prove that $q + 1 \neq…","stack exchange open-problems tag (mathoverflow, score 7, 1 answers)",,https://mathoverflow.net/questions/230174/if-n-qn2-is-an-odd-perfect-number-with-gcdq-n-1-is-it-possible-to-hav,
se-mo-231922,open,compute-checkable-small-cases,mathematics / algebraic topology,"Torsion-freeness of two groups with 2 generators and 3 relators and Kaplansky Zero Divisor Conjecture. Let $G_1$ and $G_2$ be the groups with the following presentations: $$G_1=\langle a,b \;|\; (ab)^2=a^{-1}ba^{-1}, (a^{-1}ba^{-1})^2=b^{-2}a, (ba^{-1})^2=a^{-2}b^2 \rangle,$$ $$G_2=\langle a,b \;|\; ab=(a^{-1}ba^{-1})^2, (b^{-1}ab^{-1})^2=a^{-2}b, (ba^{-1})^2=a^{-2}b^2 \rangle,$$ Are these groups torsion-free? Motivation: In both of these groups…","stack exchange open-problems tag (mathoverflow, score 13, 2 answers)",,https://mathoverflow.net/questions/231922/torsion-freeness-of-two-groups-with-2-generators-and-3-relators-and-kaplansky-ze,
se-mo-2323,open,compute-checkable-small-cases,mathematics / symplectic geometry,"Hamiltonian $S^1$ actions with isolated fixed points. I have in mind the following question for some time. Is there an example of a compact symplectic manifold with a Hamiltonian $S^1$ -action with isolated fixed points, that does not admit a compatible $S^1$ -invariant Kahler strucutre? One would say, of course there should be such an example. But I have not seen any... Added, 2009. Apparently this…","stack exchange open-problems tag (mathoverflow, score 12, 3 answers)",,https://mathoverflow.net/questions/2323/hamiltonian-s1-actions-with-isolated-fixed-points,
se-mo-2328,open,compute-checkable-small-cases,mathematics / noncommutative algebra,"When does the converse to Schur's Lemma hold? Let $R$ be a commutative ring, let $A$ be an $R$-algebra, and let $M$ be an $A$-module. If $M$ is simple, then End$_{A-mod}(M)$ is a division ring. A common use is when $R$ is the complex numbers $\mathbb{C}$, and $M$ is such that End$_{A-mod}(M)$ is finite dimensional. Then End$_{A-mod}(M) = \mathbb{C}$. Under what circumstances (regarding $R$…","stack exchange open-problems tag (mathoverflow, score 26, 3 answers)",,https://mathoverflow.net/questions/2328/when-does-the-converse-to-schurs-lemma-hold,
se-mo-23547,open,needs-theory,mathematics / number theory,"Does pi contain 1000 consecutive zeroes (in base 10)? The motivation for this question comes from the novel Contact by Carl Sagan. Actually, I haven't read the book myself. However, I heard that one of the characters (possibly one of those aliens at the end) says that if humans compute enough digits of $\pi$, they will discover that after some point there is nothing but zeroes for a really long time.…","stack exchange open-problems tag (mathoverflow, score 48, 5 answers)",,https://mathoverflow.net/questions/23547/does-pi-contain-1000-consecutive-zeroes-in-base-10,
se-mo-235800,open,compute-checkable-small-cases,mathematics / analysis of pdes,"A long-lasting conjecture of Pólya & Szegő. There is a conjecture by Pólya & Szegő (~1950, stated in p. 159 of their book Isoperimetric Inequalities in Mathematical Physics ) which is as follows: ""Of all $n$ -gons of a fixed area, the regular $n$ -gon minimizes the first Dirichlet eigenvalue."" Surprisingly, this is still open (to my knowledge) for the general case. The only settled cases…","stack exchange open-problems tag (mathoverflow, score 34, 2 answers)",,https://mathoverflow.net/questions/235800/a-long-lasting-conjecture-of-p%c3%b3lya-szeg%c5%91,
se-mo-2369,open,compute-checkable-small-cases,mathematics / functional analysis,"In a Banach algebra, do ab and ba have almost the same exponential spectrum? Let $A$ be a complex Banach algebra with identity 1. Define the exponential spectrum $e(x)$ of an element $x\in A$ by $$e(x)= \{\lambda\in\mathbb{C}: x-\lambda1 \notin G_1(A)\},$$ where $G_1(A)$ is the connected component of the group of invertibles $G(A)$ that contains the identity. Is it true that $e(ab)\cup\{0\} = e(ba)\cup\{0\}$ for all $a,b…","stack exchange open-problems tag (mathoverflow, score 21, 2 answers)",,https://mathoverflow.net/questions/2369/in-a-banach-algebra-do-ab-and-ba-have-almost-the-same-exponential-spectrum,
se-mo-23859,open,compute-checkable-small-cases,mathematics / riemann surfaces,"Covers of Riemann surfaces which become arbitrary close in Teichmuller space. Suppose $S$ and $S'$ are two compact Riemann surfaces of genus $g$. Does there exist a sequence of genera $g_i \to \infty$ and covers $S_i, S_{i}'$ of $S,S'$, both of genus $g_i$, such that $d(S_i,S_{i}')\to 0$? Here $d$ a ""natural"" distance function on Teichmuller space, of which I suppose there are many, but for definiteness let's take it to be…","stack exchange open-problems tag (mathoverflow, score 8, 2 answers)",,https://mathoverflow.net/questions/23859/covers-of-riemann-surfaces-which-become-arbitrary-close-in-teichmuller-space,
se-mo-239671,open,needs-theory,mathematics / functional analysis,"Can we solve the FGF problem by finding an appropriate action? If we can find an action of the free group $\mathbb{F}_2$ on a measure space $X$ such that the crossed product $M=L^∞(X)⋊\mathbb{F}_2$ is a ${\rm III}_1$ factor with core isomorphic to $L(\mathbb{F}_2)⊗B(H)$, then the fundamental group of $L(\mathbb{F}_2)$ is $\mathbb{R}_+$, and the free group factors isomorphism problem is solved. Remark : There…","stack exchange open-problems tag (mathoverflow, score 4, 1 answers)",,https://mathoverflow.net/questions/239671/can-we-solve-the-fgf-problem-by-finding-an-appropriate-action,
se-mo-24144,open,compute-checkable-small-cases,mathematics / general topology,"Is there an uncountable, non-discrete, Hausdorff Toronto space? We call a topological space $X$ a Toronto space if for any subspace $Y \subseteq X$ such that $Y$ and $X$ have the same cardinality it follows that $Y$ is homeomorphic to $X$. Does anybody know what is known about the following question?: Is there an uncountable, non-discrete, Hausdorff Toronto space? It is not hard to show that if $X$ is…","stack exchange open-problems tag (mathoverflow, score 11, 2 answers)",,https://mathoverflow.net/questions/24144/is-there-an-uncountable-non-discrete-hausdorff-toronto-space,
se-mo-24281,open,needs-theory,mathematics / complex variables,"Current status of Bloch constant and Landau constant bounds. The Bloch constant B (based on a theorem introduced by André Bloch in 1925 on the maximum radius of a one-to-one disk in the image of a normalized analytic function of the unit disk, see for instance Remmert ""Funktionentheorie II"" or Steven Finch marvelous ""Mathematical constants"" ) was conjectured by Ahlfors to be $$…","stack exchange open-problems tag (mathoverflow, score 8, 1 answers)",,https://mathoverflow.net/questions/24281/current-status-of-bloch-constant-and-landau-constant-bounds,
se-mo-24352,open,compute-checkable-small-cases,mathematics / combinatorics,Cutting convex sets. Any bounded convex set of the Euclidean plane can be cut into two convex pieces of equal area and circumference. Can one cut every bounded convex set of the Euclidean plane into an arbitrary number $n$ of convex pieces having equal area and circumference? The solution of this problem for $n=2$ is generically unique. Are there other values of $n$…,"stack exchange open-problems tag (mathoverflow, score 20, 3 answers)",,https://mathoverflow.net/questions/24352/cutting-convex-sets,
se-mo-245516,open,needs-theory,mathematics / functional analysis,"Does every separable Banach space have a Markushevich–Auerbach basis? Let $X$ be a separable Banach space and $X^*$ be its dual, let $\{x_i\}$ be a sequence in $X$ with dense linear span and such that there exists a sequence $\{x_i^*\}$ in $X^*$ satisfying $x_i^*(x_j)=\delta_{i,j}$ (Kronecker delta). It is clear that $\{x_i^*\}$ is uniquely determined. Let us call $\{x_i\}$ a Markushevich–Auerbach basis if the…","stack exchange open-problems tag (mathoverflow, score 4, 0 answers)",,https://mathoverflow.net/questions/245516/does-every-separable-banach-space-have-a-markushevich-auerbach-basis,
se-mo-251145,open,compute-checkable-small-cases,mathematics / computational complexity,"What arguments do exist against defining completeness in NP using injective Karp reductions? It is crucial to use the right notion of reduction to define completeness inside NP. Different notions of completeness inside NP may have significant impact on the properties of complete languages. Initially, P vs NP problem was formulated by Cook using polynomial-time Turing reductions. Later Karp refined the notion of completeness using…","stack exchange open-problems tag (mathoverflow, score 4, 1 answers)",,https://mathoverflow.net/questions/251145/what-arguments-do-exist-against-defining-completeness-in-np-using-injective-karp,
se-mo-251996,open,compute-checkable-small-cases,mathematics / differential geometry,"Is the Gromov conjecture still open? Today I read about Gromov's definition of minimal volume for smooth manifolds. $$\min {\rm Vol}(M):=\inf_{|K_g|\leq1}\{{\rm Vol}(M,g)\}.$$ Gromov's conjecture states that for every closed simply connected odd-dimensional manifold $\min {\rm Vol}(M)=0$. Is the Gromov conjecture still open? Can anybody give an example for this conjecture? Update: My…","stack exchange open-problems tag (mathoverflow, score 16, 2 answers)",,https://mathoverflow.net/questions/251996/is-the-gromov-conjecture-still-open,
se-mo-252424,open,needs-theory,mathematics / billiards,Current state of Straus's illumination problem. In George W. Tokarsky's Polygonal Rooms Not Illuminable from Every Point (1995) it is stated that the problem Is a polygonal region illuminable from at least one point in the region? was still open at the time. What is its current state? Has it been settled or is it still,"stack exchange open-problems tag (mathoverflow, score 8, 2 answers)",,https://mathoverflow.net/questions/252424/current-state-of-strauss-illumination-problem,
se-mo-257015,open,compute-checkable-small-cases,mathematics / general mathematics,"Decidable open problems. Are there any significant open problems in mathematics which are clearly decidable (in that it is easy to write a clearly corresponding program which will eventually output either Yes or No (or whatever sort of answer, out of finitely many possibilities, is appropriate), though it may take an implausibly long time to do so) but which remain open?…","stack exchange open-problems tag (mathoverflow, score 21, 2 answers)",,https://mathoverflow.net/questions/257015/decidable-open-problems,
se-mo-26040,open,needs-theory,mathematics / combinatorics,"Ramsey multiplicity. Given a positive integer $a$ , the Ramsey number $R(a)$ is the least $n$ such that whenever the edges of the complete graph $K_n$ are colored using only two colors, we necessarily have a copy of $K_a$ with all its edges of the same color. For example, $R(3)= 6$ , which is usually stated by saying that in a party of 6 people, necessarily there are…","stack exchange open-problems tag (mathoverflow, score 23, 2 answers)",,https://mathoverflow.net/questions/26040/ramsey-multiplicity,
se-mo-26330,open,needs-theory,mathematics / geometry,"Dissecting trapezoids into triangles of equal area. [Lightly edited for copy and proper formatting of mathematics. -- Pete L. Clark] The Background: Let $T$ be a trapezoid. Sherman Stein, using valuation theory, showed that if $T$ is dissectible into n triangles of equal area, then $\frac{n}{(r+1)}$, where $r$ is the ratio of the parallel sides, is an algebraic integer. See Monsky and Jepsen,…","stack exchange open-problems tag (mathoverflow, score 10, 0 answers)",,https://mathoverflow.net/questions/26330/dissecting-trapezoids-into-triangles-of-equal-area,
se-mo-26358,open,compute-checkable-small-cases,mathematics / number theory,"Can we color Z^+ with n colors such that a, 2a, ..., na all have different colors for all a? For example for n=2 coloring odd numbers red, numbers of the form 4k+2 blue and so on works. This problem was posed in the KoMaL for n+1 prime, by Peter Pach Pal. I verified it for all n<30, I think with a computerprogram one can easily verify it for much bigger numbers by trying certain periodic colorings. Ideas. As there is a lot of discussion…","stack exchange open-problems tag (mathoverflow, score 47, 6 answers)",,https://mathoverflow.net/questions/26358/can-we-color-z-with-n-colors-such-that-a-2a-na-all-have-different-color,
se-mo-26525,open,needs-theory,mathematics / packing and covering,"Coiling Rope in a Box. What is the longest rope length L of radius r that can fit into a box? The rope is a smooth curve with a tubular neighborhood of radius r , such that the rope does not self-penetrate. For an open curve, each endpoint is surrounded by a ball of radius r . For a box of dimensions $1{\times}1{\times}\frac{1}{2}$ and rope with $r=\frac{1}{4}$, perhaps…","stack exchange open-problems tag (mathoverflow, score 21, 1 answers)",,https://mathoverflow.net/questions/26525/coiling-rope-in-a-box,
se-mo-265401,open,compute-checkable-small-cases,mathematics / differential geometry,"Surface bundles over surfaces with(out) flat structure. I vaguely remember that I once attended a seminar or conference talk in which it was mentioned that the following question is open. Is there a (smooth) surface bundle over a surface $\Sigma_h \to E \to \Sigma_g$ that does not admit a flat structure? (Equivalently, one can ask whether there is a homomorphism $\pi_1(\Sigma_g) \to \pi_0…","stack exchange open-problems tag (mathoverflow, score 4, 1 answers)",,https://mathoverflow.net/questions/265401/surface-bundles-over-surfaces-without-flat-structure,
se-mo-267936,open,compute-checkable-small-cases,mathematics / group theory,"Is the fundamental group of any compact hyperbolic 3-manifold embeddable into a p-adic group? Is it true that for every compact hyperbolic $3$-manifold $M$ there exists a prime $p$, a finite field extension $K/\mathbb{Q}_p$, and an injective group homomorphism $$\tau \colon \pi_1(M) \to \mathrm{PSL}_2(\mathcal{O}_K) ?$$ Here $\mathcal{O}_K$ is the ring of integers of $K$. This question is obviously related to the existence of faithful low…","stack exchange open-problems tag (mathoverflow, score 12, 2 answers)",,https://mathoverflow.net/questions/267936/is-the-fundamental-group-of-any-compact-hyperbolic-3-manifold-embeddable-into-a,
se-mo-26821,open,needs-theory,mathematics / group theory,"Is Thompson's Group F amenable? Last year a paper on the arXiv (Akhmedov) claimed that Thompson's group $F$ is not amenable, while another paper, published in the journal ""Infinite dimensional analysis, quantum probability, and related topics"" (vol. 12, p173-191) by Shavgulidze claimed the exact opposite, that $F$ is amenable. Although the question of which, if either, was a…","stack exchange open-problems tag (mathoverflow, score 68, 7 answers)",,https://mathoverflow.net/questions/26821/is-thompsons-group-f-amenable,
se-mo-270616,open,needs-theory,mathematics / general topology,"Open bilinear maps that are not uniformly open. A map $f\colon X\to Y$ between metric spaces is uniformly open whenever for each $\varepsilon >0$ there is $\delta >0$ such that for any $x\in X$ one has $$B_Y\big(f(x),\delta\big)\subseteq f\big(B_X(x,\varepsilon)\big), $$ where $B_X, B_Y$ denote open balls in the respective spaces. Uniformly open maps have the property that they are surjective…","stack exchange open-problems tag (mathoverflow, score 15, 1 answers)",,https://mathoverflow.net/questions/270616/open-bilinear-maps-that-are-not-uniformly-open,
se-mo-272386,open,needs-theory,mathematics / mathematical physics,Random N-body problem. Suppose there are $N$ unit-mass particles whose initial positions are uniformly distributed in a unit-radius disk. Each particle is assigned a randomly oriented initial velocity vector $v_i$ of length $1$ (green below). Added : Robert Israel's incisive comment suggests that I should also stipulate that $\sum_i v_i = 0$. Then the particles act upon…,"stack exchange open-problems tag (mathoverflow, score 13, 3 answers)",,https://mathoverflow.net/questions/272386/random-n-body-problem,
se-mo-275169,open,compute-checkable-small-cases,mathematics / combinatorics,"Current state of the Komlos conjecture on vector balancing. Komlos Conjecture : the exists an absolute constant $K>0$ such that for all $d$ and any collection of vectors $v_1,\ldots, v_n\in \mathbb{R}^d$ with $\left\lVert v_i\right\rVert _2=1$ we can find weights $w_i\in\{-1,1\}$ such that $\left\lVert w_1v_1+\cdots+w_nv_n\right\rVert _{\infty}<K$. I would like to ask what the current state of this…","stack exchange open-problems tag (mathoverflow, score 20, 3 answers)",,https://mathoverflow.net/questions/275169/current-state-of-the-komlos-conjecture-on-vector-balancing,
se-mo-277495,open,compute-checkable-small-cases,mathematics / differential geometry,Limit cycles as closed geodesics(2). Hilbert 16th problem asks for a uniform upper bound $H(n)$ for the number of limit cycles of a polynomial vector field of degree $n$ on the plane. Here is an updated proof of the finitness part of the Hilbert 16th problem: http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=8352&option_lang=eng This problem is open even for $n=2$…,"stack exchange open-problems tag (mathoverflow, score 7, 0 answers)",,https://mathoverflow.net/questions/277495/limit-cycles-as-closed-geodesics2,
se-mo-27851,open,needs-theory,mathematics / polynomials,"Polynomials having a common root with their derivatives. Here is a question someone asked me a couple of years ago. I remember having spent a day or two thinking about it but did not manage to solve it. This may be an open problem, in which case I'd be interested to know the status of it. Let $f$ be a one variable complex polynomial. Supposing $f$ has a common root with every $f^{(i)},i=1,\ldots,\deg…","stack exchange open-problems tag (mathoverflow, score 59, 2 answers)",,https://mathoverflow.net/questions/27851/polynomials-having-a-common-root-with-their-derivatives,
se-mo-27861,open,compute-checkable-small-cases,mathematics / group theory,"Periodic Automorphism Towers. In Scott's classic textbook on Group Theory, he asks: Suppose that $G$ is a finite group. Is the sequence of isomorphism types of the groups $Aut^{(n)}(G)$ for $n \in \mathbb{N}$ eventually periodic? Here $Aut^{(2)}(G) = Aut(Aut(G))$ etc. Equivalently, is the sequence $|Aut^{(n)}(G)|$ always bounded above? It apparently remains opens whether the…","stack exchange open-problems tag (mathoverflow, score 26, 3 answers)",,https://mathoverflow.net/questions/27861/periodic-automorphism-towers,
se-mo-27944,open,needs-theory,mathematics / combinatorial game theory,"Do there exist chess positions that require exponentially many moves to reach? By ""chess"" here I mean chess played on an $n\times n$ board with an unbounded number of (non-king) pieces. Some care is needed if you want to generalize some of the subtler rules of chess to an $n\times n$ board, but I will not dwell on this point because the answer to the question I'm interested in should be the same under any reasonable…","stack exchange open-problems tag (mathoverflow, score 53, 4 answers)",,https://mathoverflow.net/questions/27944/do-there-exist-chess-positions-that-require-exponentially-many-moves-to-reach,
se-mo-280192,open,compute-checkable-small-cases,mathematics / number theory,"Is there always one integer between these two rational numbers? It appears that for each integer $k\geq2$, there is always one integer $c$ that satisfies the inequalities below. Can this be proved? $$\frac{3^k-2^k}{2^k-1}<c\leq \frac{3^k-1}{2^k}.$$ Note that for $k\geq2$ the lower bound is always a proper fraction and will never match an integer. Edit 28/1/2018 I have a short proof here on Overleaf. <--- It's…","stack exchange open-problems tag (mathoverflow, score 9, 3 answers)",,https://mathoverflow.net/questions/280192/is-there-always-one-integer-between-these-two-rational-numbers,
se-mo-281356,open,needs-theory,mathematics / group theory,Does the image of the exponential map generate the group? Let $G$ be a connected Fréchet-Lie group and let $\mathfrak g$ be its Lie algebra. Does the image $\exp(\mathfrak g) \subset G$ of the exponential map generate,"stack exchange open-problems tag (mathoverflow, score 16, 1 answers)",,https://mathoverflow.net/questions/281356/does-the-image-of-the-exponential-map-generate-the-group,
se-mo-282596,open,definition-dispute,mathematics / real analysis,"Characterisation of bell-shaped functions. This is an open problem that I learned from Thomas Simon. I will completely understand if the question is judged as non-research level (and it is indeed not related to my research), but I believe a solution would result in some nice, publishable mathematics. The point of this post is to popularise the problem. We need two definitions. Definition…","stack exchange open-problems tag (mathoverflow, score 18, 2 answers)",,https://mathoverflow.net/questions/282596/characterisation-of-bell-shaped-functions,
se-mo-283109,open,needs-theory,mathematics / differential geometry,"Converse of the Archimedean property of the sphere. In his remarkable book On the Sphere and Cylinder , where he came tantalizingly close to discovering calculus, Archimedes showed that the area of the portion of the sphere contained between a pair of parallel planes cutting the surface depends only on the distance between the planes (see p. 625 of this paper of King ). This fact, which has been…","stack exchange open-problems tag (mathoverflow, score 42, 0 answers)",,https://mathoverflow.net/questions/283109/converse-of-the-archimedean-property-of-the-sphere,
se-mo-283370,open,needs-theory,mathematics / differential geometry,"Plank invariant measures on convex bodies. Let $K\subset R^2$ be a convex body , i.e., a compact convex set with interior points. A plank $P$ is the region between a pair of parallel lines in $R^2$. Let us say that $P$ intersects $K$ properly provided that both boundary lines of $P$ intersect $K$. Question: Let $K\subset R^2$ be a convex body with diameter $d$. Suppose there exists a…","stack exchange open-problems tag (mathoverflow, score 10, 0 answers)",,https://mathoverflow.net/questions/283370/plank-invariant-measures-on-convex-bodies,
se-mo-283476,open,needs-theory,mathematics / differential geometry,"Converse to Euclid's fifth postulate. There is a fascinating open problem in Riemannian Geometry which I would like to advertise here because I do not think that it is as well-known as it deserves to be. Euclid's famous fifth postulate, or more precisely Playfair's version of it, states that, in the Euclidean plane, through every point outside a line $\ell$ there passes one and only…","stack exchange open-problems tag (mathoverflow, score 81, 1 answers)",,https://mathoverflow.net/questions/283476/converse-to-euclids-fifth-postulate,
se-mo-283568,open,needs-theory,mathematics / differential geometry,"Intrinsic vs Extrinsic geometry of convex surfaces. By Alexandrov's isometric embedding theorem, any locally convex metric prescribed on the sphere admits a realization as a convex surface in Euclidean 3-space, which, by Pogorelov's rigidity result, is unique up to an isometry of the ambient space. Thus, the intrinsic geometry of a convex surface completely determines its extrinsic geometry, and in…","stack exchange open-problems tag (mathoverflow, score 13, 2 answers)",,https://mathoverflow.net/questions/283568/intrinsic-vs-extrinsic-geometry-of-convex-surfaces,
se-mo-283921,open,compute-checkable-small-cases,mathematics / combinatorics,"Equilaterally triangulated surfaces with prescribed boundary. There is a problem in Richard Kenyon's list ( Wayback Machine ) which I would like to post here, because although I have thought about it from time to time, I have not been able to make the slightest progress on it: Question: ""Given a closed polygonal path $p$ in $R^3$ composed of unit segments, is there an immersed polygonal surface whose faces…","stack exchange open-problems tag (mathoverflow, score 22, 3 answers)",,https://mathoverflow.net/questions/283921/equilaterally-triangulated-surfaces-with-prescribed-boundary,
se-mo-284612,open,compute-checkable-small-cases,mathematics / euclidean geometry,Can one find a Jordan curve which has exactly one inscribed rectangle? In On the number of inscribed squares of a simple closed curve in the plane it is shown that Theorem: For every positive integer $n$ there is a simple closed curve in the plane (which can be taken infinitely differentiable and convex) which has exactly $n$ inscribed squares. this is my question: Can one find a Jordan curve which has exactly one…,"stack exchange open-problems tag (mathoverflow, score 3, 1 answers)",,https://mathoverflow.net/questions/284612/can-one-find-a-jordan-curve-which-has-exactly-one-inscribed-rectangle,
se-mo-284876,open,needs-theory,mathematics / banach spaces,"On statistical bases in Banach spaces. Let $K$ be a subset of the positive integers $\mathbb{N}$. For each $n\in \mathbb{N}$, $K_{n}$ denotes the set $\{k\in K: k\leq n\}$ and $|K_{n}|$ denotes the number of the elements in $K_{n}$. The natural density of $K$ is defined by $$\delta(K)=\lim_{n\rightarrow \infty}\frac{|K_{n}|}{n}.$$ A sequence $(x_{k})_{k}$ in a Banach space $X$ is said…","stack exchange open-problems tag (mathoverflow, score 9, 2 answers)",,https://mathoverflow.net/questions/284876/on-statistical-bases-in-banach-spaces,
se-mo-284921,open,compute-checkable-small-cases,mathematics / number theory,Undecidable easy arithmetical statement. Is there a basic arithmetic statement which is known to be undecidable ? By basic arithmetic statement I do mean an easy statement in the spirit of the Collatz conjecture . By the way is there some reasons to believe that the Collatz conjecture is undecidable,"stack exchange open-problems tag (mathoverflow, score 18, 4 answers)",,https://mathoverflow.net/questions/284921/undecidable-easy-arithmetical-statement,
se-mo-284932,open,compute-checkable-small-cases,mathematics / group theory,"Portability of Thompson theorem about solvability to Moufang loops. Say we have a finite Moufang Loop $Q$ , $|Q|<\infty$ . There is a theorem proved by Thompson that states: Group $G$ , $|G|<\infty$ is solvable $\iff$ $\forall a, b \in G$ , $\langle a, b\rangle$ is solvable. My question is: can we translate the theorem to the finite Moufang loops (maybe with some extra constraints)? The statement is reasonable…","stack exchange open-problems tag (mathoverflow, score 3, 2 answers)",,https://mathoverflow.net/questions/284932/portability-of-thompson-theorem-about-solvability-to-moufang-loops,
se-mo-289148,open,needs-theory,mathematics / probability,"Asymptotic form of pdf of Escape Time of arithmetic fBm. I am trying to apply the Girsanov formula and Doobs optional sampling theorem to obtain an asymptotic form of first passage density of an fbm process with drift, but the answer i am getting seems strange, it does not match the brownian motion case result as well as i assumed it would. I am seeking help in debugging these computations. I am also…","stack exchange open-problems tag (mathoverflow, score 4, 1 answers)",,https://mathoverflow.net/questions/289148/asymptotic-form-of-pdf-of-escape-time-of-arithmetic-fbm,
se-mo-289554,open,compute-checkable-small-cases,mathematics / number theory,"If $a^3+b^3+c^3=N$, then $x^3+y^3+z^3+t^3 = N$ in infinitely many ways? It is well-known that, $$a^3+b^3+c^3 = N\tag1$$ for $N=1,\,2$ is solvable in the integers in infinitely many ways . However, it is an open question ( but is conjectured ) that if for general $N$ it has a solution, then it must have infinitely many. I'd like to propose a variant, Conjecture: If $a^3+b^3+c^3=N$, then $x^3+y^3+z^3+t^3 = N$ in…","stack exchange open-problems tag (mathoverflow, score 4, 0 answers)",,https://mathoverflow.net/questions/289554/if-a3b3c3-n-then-x3y3z3t3-n-in-infinitely-many-ways,
se-mo-2897,open,needs-theory,mathematics / finite groups,"A group action of the Heisenberg group with special symmetries. Suppose we look at the Heisenberg group $H_{d}$ as a matrix group of upper triangular matrices over the ring $\mathbb{Z}/d\mathbb{Z}$. You can even choose $d$ to be prime if you want. A natural irrep of $H_{d}$ acting on $\mathbb{C}^{d}$ maps the group elements into the ""shift"" and ""phase"" operators, plus roots of unity. More specifically, the two…","stack exchange open-problems tag (mathoverflow, score 11, 2 answers)",,https://mathoverflow.net/questions/2897/a-group-action-of-the-heisenberg-group-with-special-symmetries,
se-mo-29000,open,needs-theory,mathematics / metric geometry,"Volumes of sets of constant width in high dimensions. Background The $n$ -dimensional Euclidean ball of radius $1/2$ has width $1$ in every direction. Namely, when you consider a pair of parallel tangent hyperplanes in any direction the distance between them is $1$ . There are other sets of constant width $1$ . A famous one is the Reuleaux triangle in the plane. The isoperimetric inequality implies…","stack exchange open-problems tag (mathoverflow, score 96, 2 answers)",,https://mathoverflow.net/questions/29000/volumes-of-sets-of-constant-width-in-high-dimensions,
se-mo-291754,open,compute-checkable-small-cases,mathematics / number theory,What is the smallest sphere whose surface includes 100 integer points? Let $S(r)$ be the surface of the origin-centered sphere in $\mathbb{R}^3$. A point is an integer point if all its coordinates are integers. What is the smallest radius $r_n$ such that $S(r_n)$ includes $\ge n$ integer points? What is the growth rate of $r_n$ with respect to $n$? Is there an algorithm that could compute $r_n$ for a specific given…,"stack exchange open-problems tag (mathoverflow, score 10, 4 answers)",,https://mathoverflow.net/questions/291754/what-is-the-smallest-sphere-whose-surface-includes-100-integer-points,
se-mo-293005,open,needs-theory,mathematics / number theory,"What is known about the relationship between Fermat's last theorem and Peano Arithmetic? As far as I know, whether Fermat's Last Theorem is provable in Peano Arithmetic is an open problem. What is known about this problem? In particular, what is known about the arithmetic systems $PA + \text{Fermat's Last Theorem}$ and $PA + \lnot \text{Fermat's Last Theorem}$ ? Note in particular that if $PA \vdash \text{Fermat's Last Theorem}$ , the…","stack exchange open-problems tag (mathoverflow, score 22, 1 answers)",,https://mathoverflow.net/questions/293005/what-is-known-about-the-relationship-between-fermats-last-theorem-and-peano-ari,
se-mo-297655,open,compute-checkable-small-cases,mathematics / number theory,"Is a stronger version of the Erdős-Turan conjecture on arithmetic progessions reasonable? (And related questions.). Define the size, possibly $\infty$, of a set $S\subseteq \mathbb{N}$ as $|S|=\sum\limits_{n\in S} \frac{1}{n}$. Then the Erdős-Turan conjecture states that if $|S|=\infty$, S must contain arbitrarily long APs (arithmetic progressions). Q1 : has anyone conjectured that for each $k>0$ there must be a finite constant $c_k$ such that: $S$ is $k$-free…","stack exchange open-problems tag (mathoverflow, score 6, 0 answers)",,https://mathoverflow.net/questions/297655/is-a-stronger-version-of-the-erd%c5%91s-turan-conjecture-on-arithmetic-progessions-re,
se-mo-301135,open,compute-checkable-small-cases,mathematics / probability,"Upper bound of the expectation of sum of the absolute value pairs. We have two arrays $A,B$ of length $n$. All values are i.i.d drawn from same distribution on $[0,1]$. If we sort $A,B$ in non-decreasing order and let $A_{(i)},B_{(i)}$ denote the i-th value in the sorted array. Then we have $S=\sum_{i=1}^n|A_{(i)}-B_{(i)}|$. What is the upper bound of $\mathbb{E}[S]$? I believe it must be the uniform distribution…","stack exchange open-problems tag (mathoverflow, score 7, 4 answers)",,https://mathoverflow.net/questions/301135/upper-bound-of-the-expectation-of-sum-of-the-absolute-value-pairs,
se-mo-302598,open,needs-theory,mathematics / graph theory,"A question related to Conways 99 graph problem. I have observed that the number of triangles $\frac{vk}{6}$ of a strongly regular graph with parameters $(v,k,1,2)$ is given by the coefficient $2(k-1)$ in the molien series of the ""4-D extraspecial group $2^{1+2\cdot 2}$"": For example the beginning of the molien series is: $1 + t^2 + 5t^4 + 6t^6 + 15t^8 + 19t^{10} + 35t^{12} + 44t^{14} + 69t^{16}…","stack exchange open-problems tag (mathoverflow, score 8, 0 answers)",,https://mathoverflow.net/questions/302598/a-question-related-to-conways-99-graph-problem,
se-mo-30655,open,compute-checkable-small-cases,mathematics / differential geometry,"Birkhoff conjecture about integrable billiards. There is a conjecture by Birkhoff which claims that for a simple closed $C^2$ plane curve $C$, if the billiard ball map is integrable then the curve is an ellipse. Integrability here might be formulated as follows: there exists a neighbourhood of $C$ in the interior $Int(C)$ that is foliated by caustics (caustics being curves that are everywhere…","stack exchange open-problems tag (mathoverflow, score 9, 4 answers)",,https://mathoverflow.net/questions/30655/birkhoff-conjecture-about-integrable-billiards,
se-mo-309044,open,compute-checkable-small-cases,mathematics / number theory,"Is there a known Turing machine which halts if and only if the Collatz conjecture has a counterexample? Some of the simplest and most interesting unproved conjectures in mathematics are Goldbach's conjecture, the Riemann hypothesis, and the Collatz conjecture. Goldbach's conjecture asserts that every even number greater than or equal to 4 can be written as the sum of two prime numbers. It's pretty straightforward how to create a computer program…","stack exchange open-problems tag (mathoverflow, score 41, 2 answers)",,https://mathoverflow.net/questions/309044/is-there-a-known-turing-machine-which-halts-if-and-only-if-the-collatz-conjectur,
se-mo-309496,open,compute-checkable-small-cases,mathematics / algebraic geometry,"A non-rational variety with a full exceptional collection? Does there exist a smooth non-rational projective variety whose bounded derived category of coherent sheaves admits a full exceptional collection? I could not find any examples in the literature (for instance projective spaces and intersections of quadrics, which admit full exceptional collections, are","stack exchange open-problems tag (mathoverflow, score 3, 1 answers)",,https://mathoverflow.net/questions/309496/a-non-rational-variety-with-a-full-exceptional-collection,
se-mo-310718,open,needs-theory,mathematics / dynamical systems,Updated background on Hilbert 16th problem? What is the current situation of the second part of the Hilbert 16th problem ? What are the most updated news on this,"stack exchange open-problems tag (mathoverflow, score 5, 1 answers)",,https://mathoverflow.net/questions/310718/updated-background-on-hilbert-16th-problem,
se-mo-311280,open,definition-dispute,mathematics / definitions,"What is the definition of the function T used in Atiyah's attempted proof of the Riemann Hypothesis? In Michael Atiyah's paper purportedly proving the Riemann hypothesis, he relies heavily on the properties of a certain function $T(s)$ , known as the Todd function. My question is, what is the definition of $T(s)$ ? Atiyah states that this function is defined in his paper ""The Fine Structure Constant"", but I can't seem to find a copy of the paper.…","stack exchange open-problems tag (mathoverflow, score 107, 2 answers)",,https://mathoverflow.net/questions/311280/what-is-the-definition-of-the-function-t-used-in-atiyahs-attempted-proof-of-the,
se-mo-31315,open,compute-checkable-small-cases,mathematics / metric geometry,"Is Lebesgue's ""universal covering"" problem still open? The following problem has been attributed to Lebesgue. Let ""set"" denote any subset of the Euclidean plane. What is the greatest lower bound of the diameter of any set which contains a subset congruent to every set of diameter 1? There are a number of interesting geometric problems of this type. Is it possible that some of them may be difficult to…","stack exchange open-problems tag (mathoverflow, score 49, 5 answers)",,https://mathoverflow.net/questions/31315/is-lebesgues-universal-covering-problem-still-open,
se-mo-313313,open,needs-theory,mathematics / classical analysis and odes,"Integral equality of 1st intrinsic volume of spheroid. Computations suggest that $$\int_{0}^{\infty}\int_{0}^{\infty} \sqrt{x+y^2} \cdot e^{-\frac{1}{2}(\frac{x}{s}+s^2y^2)}dxdy=\frac{2}{s}+\frac{2s^2\arctan(\sqrt{s^3-1})}{\sqrt{s^3-1}}.$$ The question is how to prove this equality. Background: this is the mean width of an ellipsoid with semiaxes $s,s,1/s^2$ and this almost completes the proof of the…","stack exchange open-problems tag (mathoverflow, score 4, 2 answers)",,https://mathoverflow.net/questions/313313/integral-equality-of-1st-intrinsic-volume-of-spheroid,
se-mo-31406,open,needs-theory,mathematics / number theory,Does listing the prime factors always stop? Take a natural number's prime factors and list them increasingly and repeating them according to multiplicity. Concatenate their decimal (or in any base) representation to get a new number and repeat the process. Does this always end in a prime number for any,"stack exchange open-problems tag (mathoverflow, score 15, 3 answers)",,https://mathoverflow.net/questions/31406/does-listing-the-prime-factors-always-stop,
se-mo-31482,open,needs-theory,mathematics / combinatorics,"The ""sensitivity"" of 2-colorings of the d-dimensional integer lattice. Consider the $d$-dimensional integer lattice, $Z^d$. Call two points in $Z^d$ ""neighbors"" if their Euclidean distance is 1 (i.e., if they differ by 1 on exactly one coordinate). Let $C$ be a two-coloring of $Z^d$, which makes each point either red or blue. We'll assume $C$ has the following ""nontriviality"" property: the origin is colored red, but…","stack exchange open-problems tag (mathoverflow, score 55, 10 answers)",,https://mathoverflow.net/questions/31482/the-sensitivity-of-2-colorings-of-the-d-dimensional-integer-lattice,
se-mo-314887,open,needs-theory,mathematics / set theory,"Is it still an open problem whether $\mathbb{R}^\omega$ is normal in the box topology? On page 205 of his Topology textbook, James Munkres made an interesting remark: It is not known whether $\mathbb{R}^\omega$ is normal in the box topology. Mary-Ellen Rudin has shown that the answer is affirmative if one assumes the continuum hypothesis. That's a reference to this paper by Mary Ellen Rudin. However, both Munkres and Rudin were…","stack exchange open-problems tag (mathoverflow, score 35, 1 answers)",,https://mathoverflow.net/questions/314887/is-it-still-an-open-problem-whether-mathbbr-omega-is-normal-in-the-box-top,
se-mo-315051,open,needs-theory,mathematics / measure theory,"Is it known how the Sigma Algebra generated by Jordan measurable sets compares to universally measurable sets and analytic sets? Unlike the collection $L$ of Lebesgue measurable sets, the collection $J$ of Jordan measurable sets do not form a Sigma algebra. (A set is Jordan measurable if and only if its characteristic function is Riemann integrable, and the characteristic function of a singleton is Riemann integrable, but the characteristic function of the rational numbers…","stack exchange open-problems tag (mathoverflow, score 16, 2 answers)",,https://mathoverflow.net/questions/315051/is-it-known-how-the-sigma-algebra-generated-by-jordan-measurable-sets-compares-t,
se-mo-315664,open,needs-theory,mathematics / real analysis,"For what sets does the Lebesgue Differentiation Theorem hold in one dimension? Lebesgue's differentiation theorem states that if $x$ is a point in $\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lebesgue integrable function, then the limit of $\frac{\int_B f d\lambda}{\lambda(B)}$ over all balls $B$ centered at $x$ as the diameter of $B$ goes to $0$ is equal almost everywhere to $f(x)$ . But if you replace…","stack exchange open-problems tag (mathoverflow, score 9, 4 answers)",,https://mathoverflow.net/questions/315664/for-what-sets-does-the-lebesgue-differentiation-theorem-hold-in-one-dimension,
se-mo-316543,open,needs-theory,mathematics / sobolev spaces,"An open problem in Sobolev spaces. Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. Suppose that there there is a bounded extension operator $$ E:W^{1,p}(\Omega)\to W^{1,p}(\mathbb{R}^n) \quad \text{and} \quad E:W^{1,q}(\Omega)\to W^{1,q}(\mathbb{R}^n), $$ where $1\leq p<q\leq\infty$ (or $1<p<q<\infty$ if you find it easier). Open problem. Does it follow that $$…","stack exchange open-problems tag (mathoverflow, score 9, 0 answers)",,https://mathoverflow.net/questions/316543/an-open-problem-in-sobolev-spaces,
se-mo-31920,open,needs-theory,mathematics / differential geometry,Curves of constant curvature on S^2. Most probably this is a well known question. Consider $S^2$ with a Riemannian metric. I would like to ask what is known about the structure of the set of simple (without self-intersections) closed curves on it of constant geodesic curvature. Here is a series of questions. 1) Is this true that through each point of $S^2$ passes a simple closed…,"stack exchange open-problems tag (mathoverflow, score 22, 3 answers)",,https://mathoverflow.net/questions/31920/curves-of-constant-curvature-on-s2,
se-mo-32226,open,compute-checkable-small-cases,mathematics / continued fractions,"Lower bounds (or less) for the period of sqrt(D) and related sequences. This is a continuation of Lower bounds for period length of continued fraction of square root which is a continuation of Upper bound of period length of continued fraction representation of very composite number square root . Yesterday, I came across a nice article with many problems on continued fractions of Michel Mendes France (from 1993):…","stack exchange open-problems tag (mathoverflow, score 8, 3 answers)",,https://mathoverflow.net/questions/32226/lower-bounds-or-less-for-the-period-of-sqrtd-and-related-sequences,
se-mo-322598,open,needs-theory,mathematics / combinatorics,Open questions about posets. Partially ordered sets ( posets ) are important objects in combinatorics (with basic connections to extremal combinatorics and to algebraic combinatorics) and also in other areas of mathematics. They are also related to sorting and to other questions in the theory of computing. I am asking for a list of open questions and conjectures about,"stack exchange open-problems tag (mathoverflow, score 37, 15 answers)",,https://mathoverflow.net/questions/322598/open-questions-about-posets,
se-mo-32412,open,compute-checkable-small-cases,mathematics / analytic number theory,"Question on consecutive integers with similar prime factorizations. Suppose that $n=\prod_{i=1}^{k} p_i^{e_i}$ and $m=\prod_{i=1}^{l} q_i^{f_i}$ are prime factorizations of two positive integers $n$ and $m$ , with the primes permuted so that $e_1 \le e_2 \cdots \le e_k$ , and $f_1 \le f_2 \le \cdots \le f_l$ . Then if $k=l$ and $e_i=f_i$ for all $i$ , we say that $n$ and $m$ are factorially equivalent . In other…","stack exchange open-problems tag (mathoverflow, score 26, 7 answers)",,https://mathoverflow.net/questions/32412/question-on-consecutive-integers-with-similar-prime-factorizations,
se-mo-324214,open,needs-theory,mathematics / functional analysis,"Open problems in Sobolev spaces. What are the open problems in the theory of Sobolev spaces? I would like to see problems that are yes or no only. Also I would like to see problems with the statements that are short and easy to understand for someone who has a basic knowledge in the theory, say at the level of the book by Evans and Gariepy. The problems do not have to be a well…","stack exchange open-problems tag (mathoverflow, score 32, 2 answers)",,https://mathoverflow.net/questions/324214/open-problems-in-sobolev-spaces,
se-mo-32477,open,compute-checkable-small-cases,mathematics / metric geometry,"Smallest area shape that covers all unit length curve. On a euclidean plane, what is the minimal area shape S, such that for every unit length curve, a translation and a rotation of S can cover the curve. What are the bounds of the shape's area if this is a open problem? When I asked this problem few years ago, someone told me it's open. I don't know if this is still open and I can't find any…","stack exchange open-problems tag (mathoverflow, score 16, 6 answers)",,https://mathoverflow.net/questions/32477/smallest-area-shape-that-covers-all-unit-length-curve,
se-mo-32785,open,needs-theory,mathematics / functional analysis,"Cyl(E) = Borel(E) for E non-reflexive Grothendieck Banach space. This is sort of a follow-up to Borel(X) = \sigma(X') for X non-separable PROBLEM: Given a Banach space $E$ over $\mathbb{K} \in \{\mathbb{C}, \mathbb{R}\}$ that has the Grothendieck property. Does $\hat{C}(E) = \mathcal{B}(E)$ imply $E$ is reflexive? (This would in turn imply that $E$ is separable). Some definitions: A Banach space is a…","stack exchange open-problems tag (mathoverflow, score 3, 1 answers)",,https://mathoverflow.net/questions/32785/cyle-borele-for-e-non-reflexive-grothendieck-banach-space,
se-mo-3307,open,needs-theory,mathematics / metric geometry,Can a discrete set of the plane of uniform density intersect all large triangles? Let S be a discrete subset of the Euclidean plane such that the number of points in a large disc is approximately equal to the area of the disc. Does the complement of S necessarily contain triangles of arbitrarily large area? (Added in July 2021:) I have recently learned that this is Danzer's problem . A Danzer set (…,"stack exchange open-problems tag (mathoverflow, score 43, 12 answers)",,https://mathoverflow.net/questions/3307/can-a-discrete-set-of-the-plane-of-uniform-density-intersect-all-large-triangles,
se-mo-3332,open,needs-theory,mathematics / general topology,"Two commuting mappings in the disk. Suppose that $f$ and $g$ are two commuting continuous mappings from the closed unit disk (or, if you prefer, the closed unit ball in $R^n$) to itself. Does there always exist a point $x$ such that $f(x)=g(x)$? If one of the mappings is invertible, then it is just a restatement of Brouwer's fixed point theorem but I do not know the answer in the…","stack exchange open-problems tag (mathoverflow, score 188, 8 answers)",,https://mathoverflow.net/questions/3332/two-commuting-mappings-in-the-disk,
se-mo-333425,open,needs-theory,mathematics / number theory,Hilbert 16th problem and dynamical Lefschetz trace formula. I would like to apply the known version of the conjectural formula (11) page 10 of the paper Number theory and dynamical Lefschetz trace formula . Disclaimer: I do not have a complete understanding of this formula but I can get an sketch of it. I just know that both sides of the formula are not number but distribution. I also know the meaning of…,"stack exchange open-problems tag (mathoverflow, score 13, 0 answers)",,https://mathoverflow.net/questions/333425/hilbert-16th-problem-and-dynamical-lefschetz-trace-formula,
se-mo-33396,open,compute-checkable-small-cases,mathematics / computational complexity,"Razborov's response to Almost Natural Proofs. This post is about Natural Proofs barrier in computational complexity. There are two recent papers related to this. They are: Amplifying lower bounds by means of self-reducibility by Eric Allender and Michal Koucký, Almost-natural proofs by Timothy Chow. Alexander Razborov mentioned them at the end of his talk in the first Barriers Workshop at IAS…","stack exchange open-problems tag (mathoverflow, score 15, 0 answers)",,https://mathoverflow.net/questions/33396/razborovs-response-to-almost-natural-proofs,
se-mo-3347,open,compute-checkable-small-cases,mathematics / number theory,"Is the set of primes ""translation-finite""? The definition in the title probably needs explaining. I should say that the question itself was an idea I had for someone else's undergraduate research project, but we decided early on it would be better for him to try adjacent and less technical questions. So it's not of importance for my own work per se, but I'd be interested to know if it…","stack exchange open-problems tag (mathoverflow, score 40, 2 answers)",,https://mathoverflow.net/questions/3347/is-the-set-of-primes-translation-finite,