Not schoolwork — the questions that resisted Erdős, Hilbert, and everyone since. The club keeps three shelves: what is still open, what AI recently settled, and what took humanity centuries.
Difficulty 5/5 — 50 shown. Clear
Berge guessed perfection means no odd holes; forty years later Chudnovsky–Robertson–Seymour–Thomas proved it in ~150 pages. The capstone of perfection.
Jaeger's summit conjecture: every bridgeless cubic graph maps into the Petersen graph — implying cycle double cover, Berge–Fulkerson, and five-flow at once.
Does fractional edge-colouring pin down integral edge-colouring up to one colour? Goldberg and Seymour guessed yes in the 1970s; Chen–Jing–Zang proved it in 2019.
Kaplansky asked whether torsion-free group rings have only trivial units. In 2021 Gardam found a counterexample over F₂ by computer search — the other conjectures survive.
Are higher signatures homotopy invariants? Novikov's 1970 conjecture, proved for vast classes via index theory, open in general.
If a hyperbolic group's boundary looks like a sphere, does the group act on hyperbolic 3-space? Cannon's 1990s conjecture, open in general.
Does the coloured Jones polynomial know the hyperbolic volume of a knot complement? Kashaev's 1997 guess, verified case by case, proved in none in general.
Can flow through porous rock focus itself into a singularity? Forced blow-up claimed September 2026 (Alpöge–Buckmaster, preprint + Lean); unforced stays open.
Can buoyancy-driven flow focus itself into a singularity? The forced case was claimed in September 2026 (Alpöge–Buckmaster, preprint + Lean); unforced stays open.
Szpiro's discriminant–conductor inequality for elliptic curves: equivalent to abc, claimed via inter-universal Teichmüller theory, but the proof is disputed and the problem stays open.
Small doubling forces approximate linear structure with polynomial — not exponential — losses. Conjectured for decades; proved by Gowers–Green–Manners–Tao in November 2023.
One of the most famous yes/no questions in geometric group theory: does Thompson's group F admit an invariant mean? Open since the 1970s, resistant from both sides.
Do high-energy quantum states on a chaotic manifold always spread out evenly, or can some cling to classical periodic orbits? Proved for arithmetic surfaces (2006); open in general.
Is the rank of the communication matrix, on a logarithmic scale, essentially the whole story of deterministic communication complexity? Open since the late 1980s.
Can a perfectly smooth ideal fluid focus itself into a singularity in finite time? Known for rough data since 2021; the smooth case would hand Navier–Stokes a blow-up blueprint.
Lars Onsager predicted in 1949 that rough fluid flows can dissipate energy without viscosity below Hölder exponent 1/3, but not above. Proved in full by 2018 via convex integration.
The permanent needs superpolynomial arithmetic circuits (VP ≠ VNP). Algebraic complexity's founding question, open since 1979.
A planar set of dimension > 1 determines a positive-measure set of distances. Conjectured threshold d/2; plane record 5/4 — Kakeya's distance cousin.
Can an algorithm decide rational solvability of Diophantine equations? Solved negatively over the integers (1970); over the rationals, wide open.
The cube minimises volume times polar volume among symmetric convex bodies. Proved through dimension 3; open from dimension 4 up.
A locally compact group acting faithfully on a manifold must be Lie. Hilbert's fifth problem, one level up — settled for Lipschitz and 3D actions, open in general.
Constant nonzero Jacobian implies global polynomial invertibility. Settled in two variables, open in three and above; stably equivalent to Dixmier's conjecture.
Preperiodic points of a degree-d map over a bounded-degree number field are uniformly bounded. The dynamical analogue of Merel's theorem — open in every degree ≥ 2.
Special L-values at s = 0 encode explicit units generating class fields. The rank-one abelian case fell recently (Dasgupta–Kakde); the full conjectures stand.
Are the free group factors on different numbers of generators isomorphic? None of the standard invariants separates them; the question has stood for decades.
Every endomorphism of the first Weyl algebra is an automorphism. Open since 1968 and stably equivalent to the Jacobian conjecture — the two fall together.
The Möbius function is orthogonal to every deterministic system of zero entropy. Open in general; implied by Chowla's conjecture and consistent with all known short-interval results.
Simultaneous primality for linear forms with no local obstruction — the linear case of Schinzel's Hypothesis H, covering twin primes and prime k-tuples. Every nontrivial case is open.
Polynomials with no local obstruction take simultaneous prime values infinitely often. The master conjecture behind twin primes, Sophie Germain primes and prime k-tuples — every nonlinear case open.
The K-theory of a reduced group C*-algebra should equal the equivariant K-homology of the classifying space for proper actions. Proven for large classes of groups; false in full generality with coefficients — the counterexamples reveal deep connections to geometric group theory.
Near the percolation threshold, cluster sizes follow power laws with universal exponents. In d=2 they are rigorously known via SLE; in d ≥ 6 mean-field exponents hold by the lace expansion. In between: not a single exponent has been proven.
Equations \(x^p\) + \(y^q\) = \(z^r\) with coprime terms and reciprocal exponents summing below 1 should have only finitely many primitive solutions across ALL exponent choices. Faltings proves finiteness per-triple; uniformity is the gap.
Every finite simple group is either cyclic of prime order, an alternating group, a Lie-type group, or one of 26 sporadic groups. ~100 authors, 500+ journal pages, spanning 50 years — the largest collaboration in pure mathematics before Polymath.
Deligne's proof of the last Weil conjecture established RH for function fields over ANY finite field — the theorem that earned his Fields Medal and revolutionised algebraic geometry via étale cohomology.
Curves of genus ≥ 2 over the rationals have only FINITELY many rational points. Faltings proved it in 1983, won the Fields Medal, and left effectiveness as the field's enduring homework.
Every elliptic curve over the rationals arises from a modular form. The full theorem — completing the bridge that felled Fermat's Last Theorem — was finished by Breuil, Conrad, Diamond and Taylor six years after Wiles' semistable case.
Counting solutions of equations over finite fields should follow deep topology: zeta functions satisfy RH-analogues because varieties behave like classical manifolds. Deligne's 1974 proof reshaped everything.
A precise promise problem about labelling cycles of pairwise consistency checks that is believed (was believed?) NP-hard — and became the load-bearing assumption for half of optimal-inapproximability theory.
γ = lim(1+1/2+...+1/n − log n) ≈ 0.5772…: we cannot even prove it is IRRATIONAL, let alone transcendental. Mathematics' most embarrassing constant.
Algebraic subvarieties should be visible in étale cohomology: which cohomology classes come from actual cycles? The arithmetic mirror of the Hodge conjecture, and nearly as stubborn.
Primes are equidistributed in arithmetic progressions on average up to modulus \(q^{1−ε}\) for EVERY ε — twice the reach of the Bombieri–Vinogradov theorem. Half the modern theory of primes leans on this doubling.
The Liouville function λ(n) should look random: correlations \sum_{n≤x} λ(n+h_1)...λ(n+h_k) should be o(x) for any fixed shifts. Two-point progress exists; three points and beyond resist.
The E₈ and Leech lattices are THE tightest packings of balls in their dimensions — proven by Maryna Viazovska\'s magic modular-form functions in a fortnight that stunned mathematics.
How wildly does the Riemann zeta function grow on the critical line? Lindelöf said barely at all — a consequence of RH that may be provable even without it.
How far apart can consecutive primes be? Cramér's probabilistic model says never much more than (log p)² — a prediction we cannot approach unconditionally.
Does every graph needing k colours contain k pairwise vertex-disjoint connected subgraphs mutually joined by edges? One conjecture that would explain ALL of colouring.
How algebraically independent can exponentials of numbers be? The master key to transcendence — prove it and Lindemann-Weierstrass, four-exponentials and much more fall out at once.
Is there a cardinality strictly between the integers and the reals? The first Hilbert problem dissolved into independence: yes in Gödel\'s universe, no in Cohen\'s — and both are legitimate.
Is the 4-sphere, up to diffeomorphism, the unique smooth closed simply-connected 4-manifold? Topology says yes; smooth structures say nobody knows — and dimension four is where smoothness goes feral.
a + b = c: how large can the radical of abc be relative to c? A statement about the deep tension between addition and multiplication — and home of mathematics' most controversial claimed proof.
