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.
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.
Find all n with n! + 1 a perfect square. Three solutions known since the 1870s-90s — (4,5), (5,11), (7,71) — and absolutely nothing since, despite searches far beyond \(10^9\).
Can \(1^k\) + \(2^k\) + ⋯ + (k−1)^k ever equal \(k^k\)? Only the trivial k=1 solution is known — and any other would need k beyond 10^(10^6).
Serve requests arriving online with k mobile servers at minimum movement cost. Can any deterministic algorithm achieve k-competitiveness against the optimal offline server placement on EVERY metric space?
Is there a point set of bounded density that intersects every convex body of volume 1? Sixty years of constructions either hit everything too sparsely or grow exponentially.
Multiply an irrational number by all powers of 2 and 3 simultaneously: must the fractional parts be dense between 0 and 1? Two commuting actions that should jointly mix — and stubbornly refuse to prove it.
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.
An angel jumps k squares per move eating tiles; a devil burns one square forever. Can the angel escape forever? Conway offered $100 — four independent proofs arrived within months in 2006.
Given a group presentation and a word, decide if the word equals the identity. Dehn asked for algorithms; logic answered that none can exist — for some groups.
Are finitely generated groups of bounded exponent necessarily finite? No — wildly no — yet their RESTRICTED cousin said yes so profoundly it earned a Fields Medal.
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 pure states extend uniquely to a bigger algebra? A quantum-measurement question from 1959, resolved in 2013 by undergraduates-will-understand polynomial inequalities — with ripple effects across engineering.
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.
How often can the SAME integer appear in Pascal's triangle? Singmaster guessed a uniform bound; the number 3003 appears six times and nobody can rule out seven.
Does a box exist with integer edges, integer face diagonals AND integer space diagonal? Three centuries of searching; not one example, no proof of impossibility.
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.
How many sets of size w force a sunflower (petals meeting pairwise in the same core)? Should be \(c^w\); sixty-four years of effort moved the base from w down to O(log w) — never to a constant.
Label any tree's vertices 1..n so that edge-differences are all distinct. Sixty years of near-misses culminated in November 2025: every large tree gets ALMOST there.
Smallest convex blanket covering EVERY set of diameter 1? Pál's 1920 regular hexagon has been shrinking for a century — most recently in 2024.
Find the smallest-area blanket that can cover a unit-length curve no matter how it bends. Sister puzzle to our solved moving sofa — and still wide open.
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.
Describe ALL abelian extensions of a number field explicitly, the way roots of unity generate them for Q. Kronecker called it his 'dream of youth'; it built class field theory and still dreams on.
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.
Bound the number of limit cycles of a polynomial vector field in terms of its degree. Over 120 years on, not even the quadratic case has a complete answer.
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.
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.
Every root of a polynomial with all roots in the unit disk should lie within distance 1 of SOME critical point. Gauss-Lucas says critical points live in the disk; Sendov asks for the finer choreography.
Can a graph drawn so that every edge meets every other edge exactly once have more edges than vertices? Conway bet $1,000 nobody finds one.
Numbers \(2^p\) − 1 that are prime power the hunt for ever-larger known primes — yet we cannot prove the supply is infinite.
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.
How few distinct distances must n plane points determine? Guth and Katz nearly closed it in 2010 — leaving only a logarithm standing between us and the answer.
The last survivor: after the three-variable case fell to a 216-character counterexample in 2026, the ORIGINAL two-variable Jacobian question remains open.
Can you always rebuild a graph from its deck of vertex-deleted cards? Ulam-style determinism for combinatorial structure — open for over eighty years.
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.
\(A^x\) + \(B^y\) = \(C^z\) with all exponents greater than 2 forces a common factor among the bases. Fermat's Last Theorem's living descendant, with a million-dollar bounty.
Perfect numbers equal the sum of their proper divisors. Euclid knew even ones come from Mersenne primes; whether an odd perfect number exists has outlived twenty-three centuries.
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.
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.
