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.
Berge guessed perfection means no odd holes; forty years later Chudnovsky–Robertson–Seymour–Thomas proved it in ~150 pages. The capstone of perfection.
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.
How large can the intersection of two free subgroups be? Hanna Neumann asked in 1957; Mineyev and Friedman proved the sharp bound independently in 2011.
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.
Ramanujan guessed in 1913 that x² + 7 = 2ⁿ has exactly five solutions; Nagell proved him right in 1948 via the arithmetic of Q(√−7).
Are 8 and 9 the only consecutive perfect powers? Catalan asked in 1844; Mihăilescu proved it in 2002 with a short cyclotomic argument.
Sixty years no human could prove that Robbins' weak axiom yields Boolean algebra; in 1996 the EQP prover found the fourteen-step proof alone. The first machine-proved landmark theorem.
Fifty years open: can n nearly-disjoint n-sets always be coloured with just n colours? Proved for all large n in 2021 by Kang–Kelly–Kühn–Methuku–Osthus.
Small doubling forces approximate linear structure with polynomial — not exponential — losses. Conjectured for decades; proved by Gowers–Green–Manners–Tao in November 2023.
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.
Exactly nine imaginary quadratic fields have class number one. Gauss listed them in 1801; proving his list complete took 166 years, involved a solution ignored for twenty years, and required computational verification beyond human capability.
In ANY triangle, the three adjacent angle trisectors meet pairwise at the vertices of an equilateral triangle. Discovered around 1874, published 1899 — pure Euclidean magic with no construction possible by compass alone.
\(a^b\) is transcendental whenever a is algebraic ≠ {0,1} and b is irrational algebraic. Resolves Hilbert's seventh problem: \(2^{√2}\) IS transcendental, closing a question open since Euler.
The primes contain arithmetic progressions of EVERY finite length. Euler noticed prime patterns in 1770; Green and Tao proved arbitrarily long ones exist — combining Szemerédi's theorem with a transference principle.
π(x) ~ x / log(x): the primes thin out according to the logarithmic integral. The single most consequential theorem in number theory, proven independently by Hadamard and de la Vallée Poussin using Riemann's zeta function.
Bond percolation on the square lattice has critical probability exactly 1/2. Harris proved no percolation below; Kesten closed the gap twenty-three years later — founding rigorous percolation theory.
Every high-dimensional normed space contains a subspace of dimension → ∞ that is ALMOST Euclidean. The theorem that launched asymptotic geometric analysis.
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.
Can every integer n ≥ 2 be partitioned into classes so that no class contains x, y, x+y? Croot proved yes via the circle method — a triumph of additive combinatorics.
|M(n)| < √n for all n, where M is the Mertens function? A conjecture implying RH — disproved in 1985 by computation so indirect that the first counterexample remains beyond reach even now.
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.
Are continuous transformation groups automatically differentiable — i.e., is every locally Euclidean topological group a Lie group? Yes: solved in 1952, with a beautiful twist left open in the non-Archimedean world.
Is every abelian group A with Ext(A, ℤ) = 0 free? Shelah's answer: YES and NO — provably independent of the standard axioms. Set-theoretic pluralism's second monument after CH.
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.
Does the prime-counting function ever outrun the logarithmic integral? Gauss's tables said never. Littlewood proved the lead changes hands infinitely often — and nobody knows where the FIRST flip happens beyond astronomical bounds.
Euler proved ζ(2) = π²/6 in 1735. Whether ζ(3) is rational stayed open for 243 years — until Roger Apéry announced a miraculous recurrence-driven proof at 64 years old.
Should a set admit a Fourier basis exactly when it tiles by translations? Spectacularly false in dimension ≥ 3 — Terence Tao's counterexample — while the low-dimensional wreckage keeps generating theorems.
Any dense subset of the n-dimensional tic-tac-toe board contains a whole line. Proven by the FIRST-ever Polymath project — dozens of mathematicians thinking in one comment thread.
Among all closed surfaces in 3-space, does the Clifford torus minimise bending energy (∫ H²)? Proven by min-max theory fifty years later, in work that revived geometric measure theory.
Can ONE single shape tile the plane, but only aperiodically? Yes — the 13-sided "hat", found in 2023 by a hobbyist mathematician and his collaborators, ending a sixty-year quest.
Is the colouring cost of a tensor product of graphs always the cheaper factor\'s? Fifty-three years of belief ended with Yaroslav Shitov\'s three-page demolition.
Must every bounded set in n-dimensional space split into n+1 pieces of smaller diameter? True in low dimensions — spectacularly false in high ones.
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.
Why is there no formula in radicals for fifth-degree equations? Abel proved none exists; Galois explained exactly why — and invented group theory doing it.
Trisect the angle, double the cube, square the circle — compass and straightedge alone. Two millennia of failure resolved by nineteenth-century algebra: all three are impossible, and the proofs founded modern field theory.
Give an algorithm deciding whether a polynomial has integer roots. Impossible — and proving impossibility wove Diophantine equations, Fibonacci numbers and computability into one of the 20th century\'s great joint theorems.
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.
Must every infinite ±1 sequence contain arbitrarily long homogeneous arithmetic progressions? Yes — Terence Tao's six-page proof ended an 83-year-old favourite, catalysed by a Polymath experiment.
S(5) = 160: the integers 1..160 can be 5-coloured with no monochromatic solution to x + y = z, but 1..161 cannot. A century-old Ramsey constant pinned by SAT.
In any tiling of space by unit cubes, must two cubes share a whole face? True up to dimension 6, false from 8 — dimension 7 held out until SAT solvers plus Lean closed it in 2020.
Can the integers be two-coloured with no monochromatic Pythagorean triple? No — and proving it required a 200-terabyte certificate, the largest proof in history at publication.
Which convex pentagons tile the plane? Exactly fifteen families — the final word delivered by Michaël Rao\'s exhaustive computer-assisted elimination.
BB(5) = 47,176,870: the maximal steps a halting 5-state Turing machine can run. Determined by a decade-long collaborative quest, machine-checked in Coq.
What is the largest area of a rigid shape that can negotiate an L-shaped corridor of unit width? Fifty-eight years of sofa-shaping ended with Jineon Baek's 119-page proof.
Every odd number greater than 5 is the sum of three primes. Two hundred seventy-one years from letter to theorem — the little sibling of our unsolved shelf's Goldbach entry.
Grocers stack oranges in hexagonal layers for good reason: Kepler said no arrangement beats face-centred cubic packing at 74.05%. Proof took 387 years and a computer.
Is a simply-connected closed 3-manifold necessarily a sphere? Ninety-nine years, one recluse, and the only Millennium Prize ever claimed.
Four colours suffice for any map. The first major theorem proved by a computer — and the first mathematical controversy about what a proof is.
No positive integers satisfy xⁿ + yⁿ = zⁿ for n > 2. Three hundred fifty-eight years, ten million pages of attempted proofs, one margin note too long to contain.
