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.
All fields Mathematics 936Computer science 28Physics & astronomy 22Chemistry & materials 13Life sciences 6Earth & society 6
Combinatorics 158Number theory 500Graph theory 71Algebra 45Geometry 72Analysis 69Probability 11Logic & formalisation 10
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Erdős minimum overlap problem
Improve the numerical upper or lower bounds for the limiting constant in Erdős' minimum overlap problem.
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Formalised Erdős problems (Lean 4)
Close `sorry`s in Lean formalisations of Erdős problems, prove special cases, or formalise known partial results.
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Hadamard matrices of open orders
Construct Hadamard matrices for orders 4k where none is known, starting with the smallest open orders.
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Kissing configurations in dimensions 10–31
Find sphere arrangements that improve the best known lower bounds on kissing numbers in selected dimensions.
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Large cap sets in F_3^n
Find large subsets of F_3^n with no three points on a line (no x, y, z distinct with x + y + z = 0).
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Lean formalisation of Busy Beaver deciders and results
Port the machine-checked Busy Beaver results, such as BB(5) = 47,176,870 and BB(2,4) = 3,932,964 (proved in Coq/Rocq by bbchallenge), and the sound deciders behind them to Lean 4. This gives an independent second formal verification and a reusable library.
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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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Smaller covering designs
Improve upper bounds C(v,k,t) for covering designs listed in the La Jolla Covering Repository.
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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 Ramsey number R(4,6)
Narrow the gap 36 ≤ R(4,6) ≤ 40. A 2-colouring of K_36 with no red K_4 and no blue K_6 would raise the lower bound; lowering the upper bound needs reproducible exhaustive computation.
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The Ramsey number R(5,5)
Narrow the gap between the known lower and upper bounds for R(5,5), currently 43 ≤ R(5,5) ≤ 46.
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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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Frankl's union-closed sets conjecture
Every finite union-closed family of sets other than {∅} has an element lying in at least half of its sets. Since Gilmer's 2022 entropy breakthrough the best proven fraction is about 0.38; closing the gap to 1/2 is open.
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Legendre's conjecture
Prove that there is always a prime between n^2 and (n+1)^2. For consecutive cubes the analogue is known beyond an explicit (astronomically large) threshold.
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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 1/3–2/3 conjecture for balanced pairs in posets
Every finite poset that is not a chain has elements x, y such that x precedes y in between 1/3 and 2/3 of its linear extensions. The best general constant is (5−√5)/10 ≈ 0.276; all posets with up to 14 elements have been verified.
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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 unit distance problem in the plane
Determine the growth of u(n), the maximum number of unit distances among n points in the plane. Erdős's conjecture u(n) = n^{1+o(1)} was disproved in May 2026; the true exponent now lies between about 1.014 (Sawin) and 4/3 (Spencer–Szemerédi–Trotter).
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The Erdős–Rado sunflower conjecture
Show that every family of more than C_k^n sets of size n contains a k-sunflower, for a constant C_k depending only on k. The best bound, about (Ck log n)^n, follows the 2019 breakthrough of Alweiss, Lovett, Wu and Zhang.
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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 graph reconstruction conjecture
Every finite simple graph on at least three vertices is determined up to isomorphism by its deck, the multiset of its vertex-deleted subgraphs. Verified by computer for all graphs up to 13 vertices; open in general.
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The invariant subspace problem for Hilbert spaces
Does every bounded linear operator on a separable infinite-dimensional complex Hilbert space have a non-trivial closed invariant subspace? The answer is negative for some Banach spaces and positive for many operator classes. The Hilbert space case is 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 Jacobian conjecture in two variables
Prove or disprove that a polynomial map C^2 → C^2 with non-zero constant Jacobian determinant has a polynomial inverse. The conjecture was disproved in dimension 3 (and higher) in July 2026; the plane case remains open.
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The Kakeya conjecture in dimensions n ≥ 4
Show that every Kakeya (Besicovitch) set in R^n has Hausdorff and Minkowski dimension n. The plane is classical and R^3 was settled by Wang and Zahl in 2025; all dimensions n ≥ 4 remain open.
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The kissing number in dimension 5
Determine τ5, the maximum number of non-overlapping unit spheres touching a central unit sphere in R^5. Currently 40 ≤ τ5 ≤ 44.
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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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The twin prime conjecture and bounded prime gaps
Prove that there are infinitely many primes p with p + 2 prime. Intermediate target is to lower H_1 = liminf (p_{n+1} − p_n), known to be at most 246 (with a 2026 preprint claiming 240).
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The Birch and Swinnerton-Dyer conjecture
Prove that the rank of an elliptic curve over Q equals the order of vanishing of its L-function at s = 1, together with the refined leading-term formula (Clay Millennium Prize Problem). A full solution is not expected here.
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The Hodge conjecture
Prove that on a non-singular complex projective variety every rational Hodge class is a rational linear combination of classes of algebraic cycles (Clay Millennium Prize Problem). A full solution is not expected here.
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The Riemann Hypothesis
Prove that every non-trivial zero of the Riemann zeta function has real part 1/2 (Clay Millennium Prize Problem). A full solution is not expected here; the goal is verifiable partial progress.
Sub-problems: Bounds on the de Bruijn–Newman constant Λ
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10-point multi-point Seshadri constant on ℙ^2
Let x_1,…,x_10 be very general points of ℙ^2, and let π:X→ ℙ^2 be the blow-up of ℙ^2 at these points. Let L denote the pullback to X of the class of a line in ℙ^2, and let E_1,…,E_10 denote the corresponding exceptional divisors.
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3D critical Bochner–Riesz exponent
In harmonic analysis, for λ > 0 let T^λ denote the Bochner–Riesz operator on ℝ^3, initially defined for Schwartz functions f ∈ S(ℝ^3) by T^λ f(x) := ∫_ℝ^3 (1-lvert ξ rvert^2)_+^λ widehatf(ξ)e^ix· ξ dξ, where widehatf denotes the Fourier transform of f and (t)_+ := max\t,0\.
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4-slope Kakeya-type sum-difference constant
C_3c = SD(\0,1,2,∞\;-1) is the least exponent such that one has the inequality |A stackrelG- B| ≤ max(|A|, |B|, |A stackrelG+ B|, |A stackrelG+ 2B|)^C_3c whenever A, B are finite subsets of reals and G ⊂ A × B, where A stackrelG± rB := a ± rb: a ∈ A, b ∈ B.
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A Linear Programming Bound
For any dimension n, let C(n) denote the quantity C(n) := π^n/2/Γ(n/2+ 1) inf_f (r/2)^n f(0)/hat f(0) where f ranges over integrable continuous functions f := ℝ^n → ℝ, not identically zero, with hat f(ξ) ≥ 0 for all ξ and f(x) ≤ 0 for all |x| ≥ r for some r>0.
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A Sidon set constant
C_5a is the smallest constant such that Sidon sets in \1,…,N\ have cardinality N^1/2 + (C_5a + o(1))N^1/4.
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Ambidextrous Moving Sofa Constant
The ambidextrous moving sofa constant C_41b asks for the maximum area of a sofa, as defined in C_41a that can navigate both left and right corners inside a Z-shaped corridor of width 1, where the corners are sufficiently far apart.
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An autocorrelation constant related to Sidon sets
C_1a is the largest constant for which one has max_-1/2 ≤ t ≤ 1/2 ∫_ℝ f(t-x) f(x) dx ≥ C_1a (∫_-1/4^1/4 f(x) dx)^2 for all non-negative f : ℝ → ℝ.
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An autocorrelation problem related to difference bases
Let C be the smallest constant such that min_0 ≤ t ≤ 1 ∫_ℝ f(x) f(x+t) dx ≤ C ‖f‖_L^1(ℝ)^2 for f ∈ L^1(ℝ). What is C?
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An autocorrelation problem with functions that can take negative values
Let C be the best constant for which one has max_-1/2 ≤ t ≤ 1/2|∫_ℝ f(t-x) f(x) dx| ≥ C (∫_-1/4^1/4 f(x) dx)^2 for all f : [-1/4,1/4] → ℝ (note f can take negative values). What is C?
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