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.
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Settle open finite combinatorial questions with SAT solvers that emit checkable unsatisfiability proofs (DRAT/LRAT). Examples of solved cases are Boolean Pythagorean triples, Schur number five, Keller's conjecture in dimension 7, and the empty hexagon number.
Find sorting networks with fewer comparators than the best known for n ≥ 13 inputs, or prove optimality.
Find a bilinear algorithm multiplying 3×3 matrices with fewer than 23 multiplications, or raise the lower bound.
Find bilinear algorithms that multiply two 4×4 matrices with fewer multiplications. The records are 48 over Q and C (2025) and 47 over GF(2) (2022).
Find better polynomial-time approximation algorithms for the metric Traveling Salesman Problem and prove the conjectured 4/3 integrality gap of the subtour LP. The best known ratio is 3/2 − ε with ε > 10^−36 (Karlin–Klein–Oveis Gharan).
Determine ω, the smallest exponent such that n×n matrices can be multiplied with n^(ω+o(1)) arithmetic operations. The best published bound is ω < 2.371339, a 2026 preprint claims ω < 2.371177, and it is conjectured that ω = 2.
In algebraic complexity theory, for each real k ≥ 0, let ω(k) denote the exponent for multiplying an n × n^k matrix by an n^k × n matrix. We define α := supk ≥ 0 : ω(k) = 2.
In the symmetric metric traveling salesman problem, one is given a complete graph K_n=(V,E) with a nonnegative symmetric cost function c:E→ ℝ_≥ 0 satisfying the triangle inequality. For S⊆ V, let δ(S) denote the set of edges with exactly one endpoint in S, and write δ(v):=δ(\v\).
Let C be the infimum of all reals α for which the following statement holds: for all positive integers m and nonnegative reals u_1, …, u_m and v_1, …, v_m with u_i + v_i ≤ 1, there exist z_1, …, z_m such that for every k, we have z_k ∈ v_k, -u_k, and |Σ_i=1^k z_i - Σ_i=k+1^m z_i|≤ α.
The integer factorization problem: Can the prime factorization of a positive integer be computed in polynomial time? We state the problem by asking if Nat.primeFactorsList is polynomial-time computable (assuming typical encodings of ℕ and List ℕ into bitstrings). Reference: Wikipedia
Strong Sensitivity Conjecture, for every Boolean function f : 0,1^n → 0,1, bs(f) ≤ s(f)^2. We call this the strong sensitivity conjecture because the original sensitivity conjecture only asked for a polynomial bound in terms of s(f).
