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.
24 shown
Build a connectome-constrained model of the C. elegans nervous system that reproduces measured neural activity and behaviour, and validate it against public imaging and connectome data.
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.
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).
Construct an exchange-correlation approximation that reaches chemical accuracy (~1 kcal/mol) across broad main-group chemistry at semi-local cost, and show it on public benchmarks.
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.
Prove that a symmetric informationally complete POVM (d^2 equiangular lines in C^d) exists in every dimension d, and extend the list of dimensions with exact or numerical solutions.
Determine reliably, with controlled numerics, where the doped two-dimensional Hubbard model (with and without next-nearest-neighbour hopping t′) is superconducting, striped or otherwise ordered.
Resolve the disagreement between lattice-QCD and data-driven (e+e− → hadrons) evaluations of the leading hadronic vacuum polarisation contribution to the muon anomalous magnetic moment.
Establish which high-pressure hydride superconductivity claims are robust and how accurately ab initio electron-phonon theory predicts their critical temperatures.
Decide whether four (or seven) mutually unbiased bases exist in C^6; only three are known, and a complete set of seven is widely believed not to exist.
Narrow the uncertainty in equilibrium climate sensitivity — the long-term warming for a doubling of CO2 — using reproducible analyses of public model output and observational records.
Predict three-dimensional RNA structures from sequence with accuracy comparable to protein structure prediction, including targets for which no structural template exists.
Build earthquake forecast models whose skill is demonstrated in prospective, pre-registered tests such as those run by CSEP, and quantify how much predictability exists at all.
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.
Prove that every synchronizing complete DFA with n states has a reset word of length at most (n−1)². The best general upper bound is about 0.1654·n³ (Shitov 2019). The conjecture has been verified by computer for small automata.
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.
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.
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.
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.
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.
Explain why local distance-ladder measurements of the Hubble constant H0 disagree with the value inferred from the cosmic microwave background under ΛCDM, or show that the disagreement is systematic.
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.
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.
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.
