How many equal spheres can kiss a central one? Solved in dimensions 1, 2, 3, 4, 8 and 24 only — every other dimension, starting with 5, remains open.
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.
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.
Each half of the interval between consecutive squares contains a prime. Stronger than Legendre's conjecture, from which it directly implies the one-prime-per-square case.
Andrica's Conjecture
open
The square roots of consecutive primes always differ by less than one. A prime-gap conjecture strictly stronger than Legendre's, open since 1986.
There is always a prime between consecutive perfect squares. The oldest open problem about prime gaps — weaker than Oppermann's, Andrica's and Cramér's conjectures, and implied by each of them.
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.
Hadwiger–Nelson covers ℝ² (between 5 and 7). What about ℝ³, ℝ⁴, and beyond? Even the growth RATE of χ(ℝ^d) as d increases is unknown within exponential factors.
\(F_0\) = 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.
Lee spheres tile ℤ^n perfectly for n ≤ 2 and diameter-specific cases. Whether perfect Lee codes exist in higher dimensions beyond known families is one of coding theory's oldest open questions.
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.
Does a projective plane of order 10 exist? No (proven by computation). Order 12? No. But whether planes exist for ALL non-prime-power orders satisfying Bruck–Ryser remains open — order 12 is the smallest unresolved case after the n=10 computation.
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.
How many self-avoiding walks of length n exist on a lattice? The growth rate (connective constant) is unknown even for the square lattice — Duminil-Copin and Smirnov solved the hexagonal case exactly in 2010, but every other lattice resists.
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.
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.
Can every balanced presentation of the trivial group be reduced to the trivial presentation by Nielsen moves plus conjugations? Connected to the smooth 4-dimensional Poincaré conjecture via handlebody calculus.
Vaught's Conjecture
open
A complete theory's countable models are either countably infinite in number or exactly continuum-many — never something in between. Model theory's deepest unresolved classification question.
Every ribbon knot bounds a singular disc with only self-intersections of one type. Does every SLICE knot (bounding a disc in 4-space) also bound such a ribbon? The first test case for distinguishing smooth from topological 4D knot theory.
Are infinitely many supersingular elliptic curves over ℚ? Kaneko–Zagier conjectured a precise count formula; the answer controls deep connections between modular forms and crystallographic groups.
The Generalised Continuum Hypothesis asks whether \(2^{ℵ_α}\) = ℵ_{α+1} at EVERY level of the cardinal hierarchy. GCH implies CH, so it inherits the same independence — but large cardinals may change the story at higher levels.
Make inscribed-square's affine cousin: does every centrally symmetric convex body contain an inscribed affine-regular hexagon? A test case for understanding symmetric structures inside asymmetric containers.
If a family of convex sets has the property that among any p sets, some q intersect, how few points pierce the entire family? The (3,2) case is the classical (p,q)-theorem; the tight bound is open for most parameters.
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 Collatz map extended to negative integers produces additional cycles beyond 0 and -1. Classifying ALL cycles of the generalised 3x±1 map is a harder cousin of the original problem.
Lehmer's Conjecture
open
Can integer polynomials have Mahler measure arbitrarily close to 1 without equalling it? Lehmer's degree-10 polynomial holds the world record ≈ 1.17628 — unbeaten since 1933.
Which closed subsets of [0,1] are invariant under both doubling and tripling mod 1? Only the trivial ones should exist — fifty-plus years of partial rigidity and the general case stands.
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).
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?
Danzer's Problem
open
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.
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.
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.
The Weil Conjectures
open
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.
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.
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.
The Tate Conjecture
open
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.