Special L-values at s = 0 encode explicit units generating class fields. The rank-one abelian case fell recently (Dasgupta–Kakde); the full conjectures stand.
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.
Randomisation achieves polylog(k)-competitiveness for k-server on every metric, independent of the space size. Best known bounds still depend on n; the k-only bound is open.
Borel's Conjecture
open
Every strong measure zero set of reals is countable. Independent of ZFC like the continuum hypothesis: consistent (Laver 1976), refuted under CH (Sierpiński 1928).
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.
Are the free group factors on different numbers of generators isomorphic? None of the standard invariants separates them; the question has stood for decades.
Can the product of two 2-spheres carry everywhere-positive curvature? Hopf's 1930s question — no example found, no obstruction proved.
Dixmier's Conjecture
open
Every endomorphism of the first Weyl algebra is an automorphism. Open since 1968 and stably equivalent to the Jacobian conjecture — the two fall together.
The Möbius function is orthogonal to every deterministic system of zero entropy. Open in general; implied by Chowla's conjecture and consistent with all known short-interval results.
The Tammes Problem
open
How evenly can N points spread on a sphere? The botanist's problem behind spherical codes — exact optima known only for scattered N, including the kissing twelve.
Dickson's Conjecture
open
Simultaneous primality for linear forms with no local obstruction — the linear case of Schinzel's Hypothesis H, covering twin primes and prime k-tuples. Every nontrivial case is open.
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.
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.
Gauss's Class Number One Problem
historic
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.
Morley's Trisector Theorem
historic
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.
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 Green–Tao Theorem
historic
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.
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.
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.
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.
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 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.
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.
The Mertens Conjecture
historic
|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.
Hilbert's Fifth Problem
historic
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.