Open problems and research questions, each a git repository: many come from curated collections and credit them. Routes are branches anyone can start or continue; results count once the checks pass.
Find a numerical semigroup with e(S)·n(S) < c(S), refuting Wilf's 1978 conjecture, or prove it; it holds for all semigroups with up to 60 gaps.
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Write 114, the smallest open case, as a sum of three integer cubes, or find a fourth representation of 3 as a sum of three cubes.
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Construct an explicit Steiner system S(t, k, n) with 6 ≤ t ≤ 9 and n < 200; no Steiner system with t > 5 has ever been exhibited.
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Disprove the smooth 4D Poincaré conjecture with two knots sharing a 0-surgery, one slice and one with a nonzero Rasmussen-type invariant.
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Prove that each of six cubic equations z² + y²z + x³ + … = 0 of size at most 24, the smallest open cases, has infinitely many integer solutions.
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Find seven distinct nonzero rationals such that the product of any two, plus one, is a rational square; sextuples are known, septuples are not.
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Find a small system of integer polynomials in five variables whose common zeros over ℕ give x₁ + 2 equal to exactly the primes.
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Find three squarefree D > 10^40 for which the class group of Q(√−D) modulo 2-torsion is (Z/n)² × Z/n³, with three different n.
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Find a non-affine polynomial in n ≥ 8 variables whose graph is a minimal hypersurface, or prove that none exists (Simon's question).
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Find a polygonal domain, possibly with holes, whose k-th Dirichlet eigenvalue lies below 4πk/area, or prove Pólya's 1954 conjecture.
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Construct, for arbitrarily large n, more than 1.8n points of the n × n grid with no three on a line; the best asymptotic family gives 1.5n.
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Find integer speeds for which no moment leaves every runner at least 1/(n+1) from the start, or prove the conjecture, now proved for n ≤ 14.
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Find an edge-weighted tree on more than six vertices whose pairwise path lengths are exactly 1, 2, …, n(n−1)/2, or prove that none exists.
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Find a nontrivial knot whose Jones polynomial equals 1, or prove that it detects the unknot; no example exists up to 24 crossings.
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Give an algorithm that, for h between 2^90 and 2^100, quickly finds an imaginary quadratic field with class number exactly h.
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Exhibit imaginary quadratic fields whose class groups have record ℓ-rank, such as 3-rank 9 or 5-rank 6, certified by independent forms.
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Find a 4-dimensional polytope with n facets whose graph has diameter greater than n − 4, or prove the Hirsch bound in dimension 4.
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Find a shape that does not tile the plane but can be surrounded by at least seven complete coronas of its own copies; the record is six.
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Find positive integers x, y with 0 < |y² − x³| < √x/100, beating the record ratio of about 46.6 in Hall's conjecture, unbroken since 1998.
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Give an explicit ribbon disk for a slice Gompf–Scharlemann–Thompson knot, a leading candidate counterexample to Fox's slice–ribbon conjecture.
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Give one explicit Hamiltonian construction that packs k ≥ 10 equal balls into a 4-ball, filling all but ε of its volume, for every ε > 0.
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Find k−1 positive k-th powers summing to a k-th power for some k ≥ 6, extending the counterexamples to Euler's conjecture for k = 4 and 5.
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Given a prime p > 2^80 and a target N with a large CM discriminant, quickly construct an elliptic curve over F_p with exactly N points.
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Find a polynomial of degree n with all n+1 coefficients nonzero integers whose square has fewer than √n nonzero coefficients.
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Find an odd composite number passing both the strong base-2 Fermat test and the strong Lucas test of Baillie–PSW; none is known below 2^64.
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