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.
0claims
0verified
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.
159 shown· page 1 of 4
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.
0claims
0verified
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.
0claims
0verified
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.
0claims
0verified
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.
0claims
0verified
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.
0claims
0verified
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.
0claims
0verified
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).
0claims
0verified
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.
0claims
0verified
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.
0claims
0verified
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.
0claims
0verified
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.
0claims
0verified
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.
0claims
0verified
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.
0claims
0verified
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.
0claims
0verified
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.
0claims
0verified
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.
0claims
0verified
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.
0claims
0verified
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.
0claims
0verified
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.
0claims
0verified
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\.
0claims
0verified
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.
0claims
0verified
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.
0claims
0verified
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.
0claims
0verified
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.
0claims
0verified
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.
0claims
0verified
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?
0claims
0verified
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?
0claims
0verified
Asymptotic counting exponent for partial Hadamard matrices
For integers n ≥ 2 and t ≥ 1, an n × t partial Hadamard matrix is a matrix with entries in \± 1\ whose rows are pairwise orthogonal. Let N_n,t denote the number of such matrices. For every fixed n one has N_n,4t = [1+o(1)] A_n,4t qquadas t → ∞, where A_n,4t := 2^4nt+(n-1)^2(8π t)^-n(n-1)/4.
0claims
0verified
Asymptotic Dobrowolski constant for Lehmer’s problem
Let α be a nonzero algebraic number of degree d, with minimal polynomial over ℤ f(X)=a_dΠ_i=1^d (X-α_i), where a_d>0 and α_1,…,α_d are the conjugates of α. Define the Mahler measure of α by M(α) := a_dΠ_i=1^d max1,lvert α_irvert.
0claims
0verified
Asymptotic essential-dimension ratio of the symmetric groups
For each integer n ≥ 1, let S_n be the symmetric group on n letters. Over a base field k, the essential dimension ed_k(S_n) is the smallest integer d such that the general degree-n polynomial x^n + a_1 x^n-1 + ⋯ + a_n can be reduced to a d-parameter form by a Tschirnhaus transformation.
0claims
0verified
Asymptotic line-count constant for smooth degree-d surfaces in ℙ^3 in characteristic 0
For each integer d ≥ 3, let ℓ_0(d) denote the maximal number of lines contained in a smooth surface of degree d in ℙ^3_ℂ. We define C_76 := limsup_d→∞ℓ_0(d)/d^2. The constant C_76 measures the quadratic growth rate of the maximal line count on smooth complex degree-d surfaces.
0claims
0verified
Beurling–Ahlfors transform constant
In harmonic analysis, the Beurling–Ahlfors transform B (also called the Ahlfors–Beurling operator) is the singular integral operator on L^p(ℂ), 1<p<∞, defined by Bf(z) = -1/π p.v.∫_ℂ f(w)/(z-w)^2 dm(w) = -1/π lim_ε→ 0^+∫_lvert w-zrvert>varepsilonf(w)/(z-w)^2 dm(w), where dm is Lebesgue measure on…
0claims
0verified
Bloch’s constant
Let D=\z∈ℂ:lvert zrvert<1\. Following standard notation, let F be the class of holomorphic functions f:D→ℂ normalized by lvert f'(0)rvert=1 (equivalently, after rotation, f'(0)=1).
0claims
0verified
Block Stacking Problem
Let n ≥ 1. Let C(n) be the largest displacement that the n^th block in a stack of identical rigid rectangular blocks of width 1 can be displaced horizontally over the edge of a table, with the stack remaining stable.
0claims
0verified
Bohnenblust–Hille constant on the Boolean cube
Degree at most d functions f:lbrace ± 1rbrace^n→ℝ have Fourier–Walsh expansion f(x)=Σ_S⊆ [n], |S|≤ d widehat f(S) x^S, x^S:=Π_i∈ Sx_i, [n]:=lbrace 1,…,nrbrace. For d∈ℕ set p_d:=2d/d+1.
0claims
0verified
Bohr radius for the bidisc
Let D^d := z=(z_1,…,z_d)∈ℂ^d: lvert z_1rvert,…,lvert z_drvert<1 be the unit polydisc, and let the Schur class S_d be the set of analytic functions f:D^dtoD.
0claims
0verified
Borcea's Conjecture
For any 1 ≤ p < ∞ and n ≥ 2, let C(p,n) be the smallest constant such that for any complex polynomial f of degree n with zeroes z_1,…,z_n satisfying 1/n Σ_i=1^n |z_i|^p ≤ 1, and every zero f(ζ)=0 of f, there exists a critical point f'(ξ) = 0 of f with |ξ - ζ| ≤ C(p,n). What is C(p,n)?
0claims
0verified
Bounded prime gap constant
Let p_n denote the n-th prime. The bounded prime gap constant is C_88a = H_1 := liminf_n → ∞ (p_n+1 - p_n), the least limit point of the sequence of gaps between consecutive primes.
0claims
0verified
Brennan's conjecture exponent
Let Ω⊂ℂ be simply connected with at least two boundary points in the extended complex plane, and let φ:ΩtoD be a conformal map. Brennan's conjecture states that ∫_Ωlvert φ'(z)rvert^p dx dy < ∞ qquadwhenever 4/3<p<4.
0claims
0verified
Brezis–Gallouet–Wainger remainder constant on the 2D torus
C_16 = L is the smallest constant for which the sharp Brezis–Gallouet inequality ‖u‖_L^∞(T^2)^2 ≤ 1/4π ‖∇ u‖_L^2(T^2)^2 Bigl[lnδ(u) + lnbigl(1+lnδ(u)bigr) + LBigr] holds for all zero-mean functions u ∈ H^2(T^2) with sufficiently large frequency ratio δ(u) := ‖Δ u‖_L^2(T^2)^2/‖∇ u‖_L^2(T^2)^2.
0claims
0verified
Brun's Constant
C_81a, Brun's Constant, is the sum of the reciprocals of the twin primes.
0claims
0verified
Burgess-quality subconvexity exponent for Dirichlet L-functions
A Dirichlet character of level q is an arithmetic function χ that is multiplicative, is defined by a character on (ℤ/qℤ)^ast on integers coprime to q, and is 0 on integers not coprime to q.
0claims
0verified
Centered Hardy–Littlewood maximal constant in dimension 2
In ℝ^d (d≥ 1), let M_d denote the centered Hardy–Littlewood maximal operator associated to cubes, defined by M_d f(x) := sup_r>0 1/lvert Q(x,r)rvert∫_Q(x,r) lvert f(y)rvert dy, where Q(x,r) is a closed ℓ_∞ ball of radius r and center x in ℝ^d, that is, a closed cube centered at x, with sides…
0claims
0verified
Chvátal–Sankoff constant for a binary alphabet
Let λ_n,2 be the random variable assigning two uniformly random binary strings of length n the length of their longest common subsequence. Then C_31a is the (well-defined) limit C_31a := lim_n → inftyE[λ_n,2]/n.
0claims
0verified
Classical zero-free region constant
C_8 = R is the least constant such that there are no zeroes σ+it of the Riemann zeta function with lvert t rvert ≥ 2 and σ > 1 - 1/R log lvert t rvert.
0claims
0verified
Constant term of one-shot channel simulation
The constant term of one-shot channel simulation [HJMR07], [BG14], [LEG18], [Li25] is given as (we use the definition in [Li25]) C_32=limsup_t→∞(sup_p_X,Y: I(X;Y)=t inf_p_S|X,Y: I(X;S)=0H(Y|S)-t-log_2t), where H(Ylvert S)=H(Y,S)-H(S) is the conditional entropy (in bits), and I(X;Y)=H(X)+H(Y)-H(X,Y)…
0claims
0verified
Conway thrackle constant
In topological graph theory, a thrackle is a drawing of a finite graph in the plane in which every pair of edges meets precisely once, either at a common endpoint or at a proper crossing.
0claims
0verified
Davenport constant for C_n^3
In zero-sum theory, the Davenport constant D(G) of a finite abelian group G is defined as the smallest integer l∈ℕ such that every sequence S over G of length lvert Srvert≥ l has a non-empty zero-sum subsequence.
0claims
0verified
de Bruin-Sharma Problem
For n ≥ 4, let Ω(n) be the set of pairs (α,β) ∈ ℝ_+^2 such that, whenever P is a degree n polynomial whose roots z_1,…,z_n sum to zero, and ξ_1,…,ξ_n-1 are the critical points (roots of P'), that |ξ_1|^4 + … + |ξ_n-1|^4 ≤ α (|z_1|^4 + … + |z_n|^4) + β (|z_1|^2 + … + |z_n|^2)^2. What is Ω(n)?
0claims
0verified
Browse by field
CombinatoricsNumber theoryGraph theoryAlgebraGeometryAnalysisProbabilityLogic & formalisationAlgorithmsComplexityComputabilityOptimisationCryptography theoryMachine learningPhysicsTheoretical physicsAstrophysics & cosmologyQuantum informationChemistryMaterialsBiologyNeuroscienceEcologyClimateEarth scienceEconomics