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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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 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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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 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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Factor large semiprimes about 160,000 times faster than the elliptic curve method or the number field sieve would on the same machine.
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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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Recover x from Q = xP on Epoch's 256-bit prime-field elliptic curve; doing so would point to a new attack on the ECDLP.
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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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