Costas arrays of order 32 and 33
Find a Costas array of order 32 or 33, the smallest orders for which none is known, or extend the complete enumeration of Costas arrays beyond order 29.
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Open problems
Each problem states how progress is verified and what counts as a contribution. Besides the problems curated here, the catalogue includes open conjectures from Formal Conjectures (with Lean statements), optimization constants and the AlphaEvolve problems. Know one that belongs here? Propose a problem.
19 shown
Optimal Golomb rulers
Find the shortest Golomb ruler (all pairwise mark differences distinct) with n marks. Optimality is proven up to 28 marks (length 585, distributed.net, 2022); 29 marks is the first open case, and shorter rulers for larger n would beat long-standing constructions.
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Small van der Waerden numbers
Determine W(r,k), the least N such that every r-colouring of {1,…,N} contains a monochromatic k-term arithmetic progression. Only seven non-trivial values are known; the open cases W(2,7), W(3,5), W(4,4) and W(5,3) invite better lower-bound colourings and exact computations.
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Snake-in-the-box — longest induced paths in hypercubes
Find the longest induced path (snake) in the n-dimensional hypercube Q_n. Optimal lengths are known only up to n = 8 (98); for n = 9–13 new records were set in 2026 and further improvements are open.
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The degree–diameter problem for graphs
Find the largest graphs with maximum degree d and diameter k. Records for 3 ≤ d ≤ 20 and 2 ≤ k ≤ 10 are tabulated and mostly far below the Moore bound; whether a Moore graph of degree 57 (3250 vertices) exists is a famous open case.
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The sixth Schur number S(6)
Find the largest N such that {1,…,N} can be split into six sum-free sets. After Heule's 2017 SAT proof that S(5) = 160, the best known bound is S(6) ≥ 536, with a large gap to the upper bound.
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Computer-assisted proofs of finite-time singularities in 3D Euler
Establish, check and extend rigorous computer-assisted proofs that smooth solutions of the 3D incompressible Euler equations (and related models) develop singularities in finite time.
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Crossing numbers of complete and complete bipartite graphs
Prove Hill's conjecture cr(K_n) = ¼⌊n/2⌋⌊(n−1)/2⌋⌊(n−2)/2⌋⌊(n−3)/2⌋ and Zarankiewicz's conjecture for K_{m,n}. Exact values are known only for small cases (K_n up to n = 14, K_{m,n} for m ≤ 6 and a few m = 7 cases).
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Do odd perfect numbers exist?
Decide whether an odd perfect number exists. Any such number exceeds 10^1500 and has at least 10 distinct prime factors; progress tightens these constraints.
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The (binary) Goldbach conjecture
Prove that every even integer greater than 2 is the sum of two primes. It has been verified up to 4·10^18, and the ternary (odd) version was proved by Helfgott.
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The chromatic number of the plane (Hadwiger–Nelson problem)
Determine how many colours are needed so that no two points of the plane at distance exactly 1 share a colour. The answer is known to be 5, 6 or 7; a concrete sub-goal is a smaller 5-chromatic unit distance graph than the 509-vertex record.
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The Collatz (3n + 1) conjecture
Prove that iterating n ↦ n/2 (n even), 3n + 1 (n odd) reaches 1 from every positive integer. It has been verified up to 2^71, and Tao showed that almost all orbits attain almost bounded values.
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The Erdős–Straus conjecture
Prove that 4/n = 1/x + 1/y + 1/z has a solution in positive integers for every n ≥ 2. It has been verified to at least 10^17, and all n outside a few residue classes are covered by explicit identities.
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The Erdős–Szekeres happy ending problem
Is every set of 2^{n−2}+1 points in general position in the plane guaranteed to contain n points in convex position? Known exactly up to n = 6 (17 points); the first open case is whether 33 points force a convex 7-gon.
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The graceful tree conjecture (Ringel–Kotzig)
Every tree with n vertices has a graceful labelling, i.e. vertex labels 0..n−1 whose edge differences are exactly 1..n−1. It has been verified for all trees with at most 35 vertices; extending this range and proving new classes graceful are open.
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The inverse Galois problem over Q
Decide whether every finite group occurs as the Galois group of a Galois extension of Q. All sporadic groups are now realised (M23 in 2026); most transitive groups of degree 24 are not yet.
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The lonely runner conjecture
For k+1 runners with distinct constant speeds on a unit circular track, each runner is at some time at distance at least 1/(k+1) from all others. Computer-assisted proofs now cover up to 13 runners; the general case is open.
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The perfect cuboid problem
Decide whether a box exists whose three edges, three face diagonals and space diagonal are all integers. Exhaustive searches show the space diagonal of any such box would exceed 2^53.
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Bounds on the de Bruijn–Newman constant Λ
Lower the known upper bound Λ ≤ 0.2 for the de Bruijn–Newman constant. The Riemann Hypothesis is equivalent to Λ = 0, and Λ ≥ 0 is known.
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