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.
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.
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.
Colour the edges of complete graphs red/blue: how large before a monochromatic K₅ is forced? For K₅ versus K₅ we know only 43 ≤ R(5,5) ≤ 48.
In any non-empty family of sets closed under union, must some element appear in at least half the sets? Perhaps the simplest open statement in extremal set theory.
What is the least exponent ω with n×n matrices multipliable in O(n^ω) operations? Fifty-six years of algebraic ingenuity brought ω from 3 down toward 2 — never reaching it.
How large can a subset of \(F_3^n\) be containing no three-term arithmetic progression? Bounds have collapsed twice — once by the polynomial method, once to DeepMind's FunSearch.
Can every fraction 4/n be written as a sum of exactly three unit fractions? Egyptian mathematics meets the distribution of prime factors.
Does every simple closed curve in the plane contain four points forming a square? Over a century old, proven true for vast classes of curves, false for none.
How many colours suffice so that no two points exactly one unit apart share a colour? Between five and seven since 2018 — when Aubrey de Grey shook a 70-year-old stalemate.
n runners start together on a circular track, each with a distinct constant speed. Must every runner, at some moment, be strictly farther than 1/n of the track from all the others?
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.
Does every bounded linear operator on a Hilbert space send some non-trivial closed subspace to itself? Fifty years of counterexamples on wilder spaces, two unrefereed claims on Hilbert space itself — and still no verdict.
How small can a set be that contains a unit line segment pointing in every direction? Dimension three fell in 2025 in a proof hailed as a once-in-a-century achievement; dimension four and beyond remain open.
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.
A divisibility question about shifted powers — the first Erdős problem genuinely resolved end-to-end by an LLM pipeline, certified by a second AI and checked by humans.
How many colours force a monochromatic triangle? Erdős's fifty-year-old lower-bound challenge, resolved superexponentially by Astra.
Are the hyperfinite II₁ factor's matrix approximations unique — must certain groups be visible in their von Neumann algebras? Disproved by OpenAI's Astra model, ending a fifty-year expectation.
Are all finitely generated groups approximable by finite ones? OpenAI's Astra model announces a construction proving not — resolving a central question of group theory.
Sums of small independent random variables resist concentrating too sharply. A decade-plus-old conjecture, initially proved by ChatGPT — and by two independent teams the same day.
Must a finite set of reals grow fast under addition or multiplication? Over the reals: no — a cascade begun by AI's unit-distance breakthrough. Over the integers the conjecture lives on.
Can every bridgeless graph be covered by circuits traversing each edge exactly twice? Yes — with at most eight loops. Settled by GPT-5.6 Sol after fifty years.
If a polynomial self-map of space has everywhere-invertible derivative, must it be globally invertible? False in three variables: a 216-character counterexample ended an 87-year hunt.
How many pairs of n plane points can sit exactly one unit apart? Erdős guessed his grid was unbeatable. An AI model proved him wrong — the first prominent open problem resolved autonomously by AI.
Give the quantum field theory behind the Standard Model a rigorous foundation, and explain why the gluon is massive despite being massless in the equations.
On algebraic varieties, the analytic topology and the algebraic geometry see the same cycles. Whether they truly align is one of geometry's deepest gaps.
An elliptic curve has infinitely many rational points precisely when its L-function vanishes at s = 1. Deep arithmetic meets deep analysis.
Do solutions of the equations that describe fluid motion always exist and stay smooth, or can they blow up? Mathematics has not caught up with the water in the glass.
pdes fluid-dynamics millennium-prize mathematical-physics blow-up regularity
There are infinitely many primes p such that p + 2 is also prime. Infinitely close, structurally — but unproven.
Every even integer greater than 2 is the sum of two primes. The oldest surviving unsolved problem in number theory.
Take any positive integer: halve it if even, else triple it and add one. Does the sequence always reach 1?
If a solution can be checked quickly, can it also be found quickly? The question that organises theoretical computer science.
All non-trivial zeros of the Riemann zeta function lie exactly on the critical line Re(s) = 1/2. The single most consequential open statement in pure mathematics.
