Jaeger's summit conjecture: every bridgeless cubic graph maps into the Petersen graph — implying cycle double cover, Berge–Fulkerson, and five-flow at once.
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.
Colouring vertices and edges together: is maximum degree plus two always enough? Behzad asked in 1965; the exact bound is still open.
Are the nth roots of the primes strictly decreasing? Firoozbakht's 1982 guess would control prime gaps far beyond Legendre — verified computationally, proved nowhere.
Are higher signatures homotopy invariants? Novikov's 1970 conjecture, proved for vast classes via index theory, open in general.
Cannon's conjecture
open
If a hyperbolic group's boundary looks like a sphere, does the group act on hyperbolic 3-space? Cannon's 1990s conjecture, open in general.
Does the coloured Jones polynomial know the hyperbolic volume of a knot complement? Kashaev's 1997 guess, verified case by case, proved in none in general.
Can flow through porous rock focus itself into a singularity? Forced blow-up claimed September 2026 (Alpöge–Buckmaster, preprint + Lean); unforced stays open.
Can buoyancy-driven flow focus itself into a singularity? The forced case was claimed in September 2026 (Alpöge–Buckmaster, preprint + Lean); unforced stays open.
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.
Iterated absolute differences of the primes always start with 1. Verified to 10^13 and beyond, proved nowhere — perhaps the most elementary open problem about the primes.
Szpiro's conjecture
open
Szpiro's discriminant–conductor inequality for elliptic curves: equivalent to abc, claimed via inter-universal Teichmüller theory, but the proof is disputed and the problem stays open.
One of the most famous yes/no questions in geometric group theory: does Thompson's group F admit an invariant mean? Open since the 1970s, resistant from both sides.
Do high-energy quantum states on a chaotic manifold always spread out evenly, or can some cling to classical periodic orbits? Proved for arithmetic surfaces (2006); open in general.
Is the rank of the communication matrix, on a logarithmic scale, essentially the whole story of deterministic communication complexity? Open since the late 1980s.
Can a perfectly smooth ideal fluid focus itself into a singularity in finite time? Known for rough data since 2021; the smooth case would hand Navier–Stokes a blow-up blueprint.
Are all hyperbolic groups residually finite? Gromov's basic question — no counterexample known, no proof in sight.
Short vectors balance to constant discrepancy with the right signs. Conjectured O(1), proved O(√log n) — the central open balancing statement.
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.
Valiant's Hypothesis
open
The permanent needs superpolynomial arithmetic circuits (VP ≠ VNP). Algebraic complexity's founding question, open since 1979.
Pillai's Conjecture
open
Gaps between perfect powers grow without bound. Catalan settled gap 1 (8 and 9); every other fixed gap is open, and abc would imply them all.
Subcomplexes of aspherical 2-complexes stay aspherical. Open since 1941; LOT complexes from ribbon discs are the test cases.
A planar set of dimension > 1 determines a positive-measure set of distances. Conjectured threshold d/2; plane record 5/4 — Kakeya's distance cousin.
Suslin's Problem
open
Does the countable chain condition characterise ℝ? Independent of ZFC: Suslin lines exist under V = L, consistently none exist (Solovay–Tennenbaum 1971).
All NP-complete problems are the same problem up to polynomial-time recoding. Open since 1977; a positive answer would reveal deep structure in NP.
Can an algorithm decide rational solvability of Diophantine equations? Solved negatively over the integers (1970); over the rationals, wide open.
How evenly can s points spread before some triangle gets tiny? Exact growth of the optimal minimal-triangle area is open between log s/s² and n^-7/6.
The cube minimises volume times polar volume among symmetric convex bodies. Proved through dimension 3; open from dimension 4 up.
The MDS Conjecture
open
How long can an optimal error-correcting code be? The MDS conjecture caps q-ary codes at length q+1 — proved for prime alphabets, open in general.
Which integers are areas of rational right triangles? Tunnell's criterion decides it assuming BSD; unconditionally, no general method is known.
Chomp is a first-player win, but nobody knows the winning first move in general. Strategy-stealing proves existence; explicit strategy unknown.
Is membership in the Mandelbrot set decidable over the reals? Penrose's 1989 question — computable in weaker models, open in Blum–Shub–Smale.
Exact van der Waerden numbers: how long an interval forces a monochromatic k-term progression? Only scattered values known; W(2,7) already open.
A polynomial sharing a factor with every derivative must be a power of a linear one. Open since 2001; settled for prime-power degrees, open in general.
A locally compact group acting faithfully on a manifold must be Lie. Hilbert's fifth problem, one level up — settled for Lipschitz and 3D actions, open in general.
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.
Below √n, no efficient algorithm finds a planted clique in a random graph. The hardness assumption behind sparse PCA, community detection and average-case crypto.
Constant nonzero Jacobian implies global polynomial invertibility. Settled in two variables, open in three and above; stably equivalent to Dixmier's conjecture.
Does BPP = P?
open
Is randomness essential to efficient computation? BPP = P would follow from strong circuit lower bounds; unconditionally, the question is wide open.
Preperiodic points of a degree-d map over a bounded-degree number field are uniformly bounded. The dynamical analogue of Merel's theorem — open in every degree ≥ 2.
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.
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.