Are the nth roots of the primes strictly decreasing? Firoozbakht's 1982 guess would control prime gaps far beyond Legendre — verified computationally, proved nowhere.
The Problems
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 3/5 — 38 shown. Clear
The Riesel problem
open
The distributed-computing classic: is 509203 the smallest k for which k·2ⁿ − 1 is never prime? PrimeGrid grinds on; the answer is a finite search away.
The Robbins conjecture
historic
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.
Is 78,557 the smallest Sierpiński number? Five candidates stand between proof and eternity; PrimeGrid is still searching.
The 1-2-3 Conjecture
open
Weight edges 1-2-3 so neighbouring vertex-sums differ. Always possible, with no isolated edges the only obstruction? Open since 2004.
Is membership in the Mandelbrot set decidable over the reals? Penrose's 1989 question — computable in weaker models, open in Blum–Shub–Smale.
Is there a dense set in the plane with all mutual distances rational? Ulam's 1946 question — the natural candidates are ruled out, the general case open.
Every connected graph except K2 has an edge-labelling with all vertex-sums distinct. Open since 1990; verified for paths, wheels, complete and dense graphs.
How many pairwise orthogonal Latin squares of order n exist? The number grows with n but the exact asymptotics connect to deep open questions about the Alon–Tarsi constant and the chromatic number of Cartesian products of complete graphs.
\(F_0\) = 3, \(F_1\) = 5, \(F_2\) = 17, \(F_3\) = 257, \(F_4\) = 65537 are all prime. Euler showed \(F_5\) is composite in 1732. Are there any more Fermat primes at all? Only five are known after nearly 400 years.
How many points in general position force a convex n-gon? Suk proved g(n) = \(2^{n+o(n)}\), matching Erdős's construction up to subexponential factors. Exact values known only for n ≤ 6.
The Gelfond–Schneider Theorem
historic
\(a^b\) 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 Prime Number Theorem
historic
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.
How many points in general position force n in convex position? Erdős named it 'the happy ending' because Szekeres and Klein met working on it — and married. The exact threshold is still unknown for n ≥ 7.
Beyond the lonely runner conjecture's worst-case bound, what is the SET of lonely times? The structure of this set — its measure, its topology — is completely open even for small runner counts.
The Erdős–Graham Conjecture
historic
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.
Brocard's Problem
open
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).
Apéry's Theorem: ζ(3) Is Irrational
historic
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.
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.
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.
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.
Moser's Worm Problem
open
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.
Borsuk's Conjecture
historic
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.
Sendov's Conjecture
open
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 you always rebuild a graph from its deck of vertex-deleted cards? Ulam-style determinism for combinatorial structure — open for over eighty years.
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.
The Fifth Schur Number S(5)
historic
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.
Which convex pentagons tile the plane? Exactly fifteen families — the final word delivered by Michaël Rao\'s exhaustive computer-assisted elimination.
Ramsey Number R(5,5)
open
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.
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.
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?
The Four Colour Theorem
historic
Four colours suffice for any map. The first major theorem proved by a computer — and the first mathematical controversy about what a proof is.
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.