# Open problems from the 1960s > 33 AI-made results on questions first raised the 1960s, each explained in plain English. Results on questions first raised the 1960s, going by the… Canonical: https://closedform.nishachay.tech/open-since/1960s/ Last updated: 2026-10-14 Results on questions first raised the 1960s, going by the earliest step each entry records. 33 AI-made results on questions first raised the 1960s, from 50 papers in the OpenAI math collection, each explained in plain English. 25 have a formal Lean proof listed; 8 are claimed in a paper and not yet machine-checked. They span 15 fields, led by theoretical computer science (6), number theory (5) and combinatorics (4). The oldest question here dates back to 1960: [A quadratic bound for Jacobsthal’s function](https://closedform.nishachay.tech/e/openai-math-2026-021/). Results rated Landmark include [The full BSD formula from low Selmer corank](https://closedform.nishachay.tech/e/openai-math-2026-002/) and [The L log L Fourier-convergence conjecture](https://closedform.nishachay.tech/e/openai-math-2026-075/). ## Results (33) - [The full BSD formula from low Selmer corank](https://closedform.nishachay.tech/e/openai-math-2026-002/) - Entry 002 · The full Birch–Swinnerton-Dyer formula is claimed for every elliptic curve with a small Selmer group at one prime. (Claimed, not checked) - [Ordinary two-point correlations and the corrected Elliott conjecture](https://closedform.nishachay.tech/e/openai-math-2026-007/) - Entry 007 · Neighboring numbers stop agreeing on prime-factor parity. (Formal proof listed) - [A quadratic bound for Jacobsthal’s function](https://closedform.nishachay.tech/e/openai-math-2026-021/) - Entry 021 · Every run of about k² whole numbers holds one sharing no factor with a number that has k prime factors. (Formal proof listed) - [Positive lower density of large prime gaps](https://closedform.nishachay.tech/e/openai-math-2026-026/) - Entry 026 · For any size bar, a fixed share of prime gaps clears it, all the way out. (Formal proof listed) - [Uniformly bounded components of Gaussian-prime graphs](https://closedform.nishachay.tech/e/openai-math-2026-028/) - Entry 028 · No walk through Gaussian primes with steps of bounded length can go on forever. (Formal proof listed) - [A characteristic-zero counterexample to Lipman–Zariski](https://closedform.nishachay.tech/e/openai-math-2026-048/) - Entry 048 · A singular complex surface whose vector fields still form a free basis, against the Lipman–Zariski conjecture. (Claimed, not checked) - [A counterexample to Griffiths’ positivity conjecture](https://closedform.nishachay.tech/e/openai-math-2026-050/) - Entry 050 · Algebraic positivity need not come with positive curvature, even on the simple surface built from two spheres. (Formal proof listed) - [The L log L Fourier-convergence conjecture](https://closedform.nishachay.tech/e/openai-math-2026-075/) - Entry 075 · Fourier sums of every L log L function converge to it at almost every point of the circle. (Formal proof listed) - [Matrix multiplication with exponent at most 9/4](https://closedform.nishachay.tech/e/openai-math-2026-107/) - Entry 107 · Huge n-by-n complex matrices can be multiplied in about n^(9/4) arithmetic steps, up to any tiny slack. (Formal proof listed) - [Integer multiplication below n log n](https://closedform.nishachay.tech/e/openai-math-2026-109/) - Entry 109 · A tape-based computer can multiply huge whole numbers slightly faster than the long-conjectured n log n limit. (Claimed, not checked) - [Almost-linear-time exact matching and prescribed-degree factors in general graphs](https://closedform.nishachay.tech/e/openai-math-2026-120/) - Entry 120 · A randomized algorithm finds a largest matching in any graph in almost-linear time, the paper claims. (Claimed, not checked) - [Exact Fourier transforms below n log n](https://closedform.nishachay.tech/e/openai-math-2026-130/) - Entry 130 · In exact arithmetic, the discrete Fourier transform can be computed in slightly fewer than n log n steps. (Formal proof listed) - [The computational complexity of Weisfeiler–Leman refinement](https://closedform.nishachay.tech/e/openai-math-2026-133/) - Entry 133 · Deciding Weisfeiler–Leman equivalence at dimension k needs time n to a power proportional to k. (Formal proof listed) - [Generalized star height at most three](https://closedform.nishachay.tech/e/openai-math-2026-134/) - Entry 134 · Three nested stars are enough to describe any regular language once complement is allowed. (Formal proof listed) - [The Harary–Hill and Zarankiewicz crossing-number formulas](https://closedform.nishachay.tech/e/openai-math-2026-165/) - Entry 165 · The classic crossing formulas for drawing complete networks are exact. (Formal proof listed) - [The circulant Hadamard and Barker-sequence conjectures](https://closedform.nishachay.tech/e/openai-math-2026-179/) - Entry 179 · Circulant Hadamard matrices exist only in sizes 1 and 4. (Formal proof listed) - [Barnette’s Hamiltonian-cycle conjecture](https://closedform.nishachay.tech/e/openai-math-2026-180/) - Entry 180 · Every Barnette graph has a round trip through all its points. (Formal proof listed) - [The Erdős–Gallai cycle-decomposition conjecture](https://closedform.nishachay.tech/e/openai-math-2026-181/) - Entry 181 · Every network on n points splits into at most a fixed multiple of n loops and single links. (Formal proof listed) - [Serre’s intersection-multiplicity conjecture](https://closedform.nishachay.tech/e/openai-math-2026-193/) - Entry 193 · Serre's intersection multiplicity is claimed to be strictly positive over every regular local ring. (Claimed, not checked) - [Lech’s multiplicity conjecture](https://closedform.nishachay.tech/e/openai-math-2026-194/) - Entry 194 · Hilbert–Samuel multiplicity is claimed never to drop along a flat local map, settling Lech's 1960 conjecture. (Formal proof listed) - [A counterexample to finitistic-dimension finiteness](https://closedform.nishachay.tech/e/openai-math-2026-198/) - Entry 198 · An algebra with finite resolutions of every length refutes the little finitistic-dimension conjecture. (Formal proof listed) - [Planar first-passage geometry and the absence of bigeodesics](https://closedform.nishachay.tech/e/openai-math-2026-212/) - Entry 212 · Planar random-metric grids are claimed to have no route that is fastest forever both ways, and a smooth limit shape. (Formal proof listed) - [Exact three- and four-state reconstruction thresholds and four-state tree capacity](https://closedform.nishachay.tech/e/openai-math-2026-229/) - Entry 229 · For three and four colors, guessing a tree's root from far leaves works exactly above the Kesten–Stigum bound. (Formal proof listed) - [Shelah's eventual categoricity and the prescribed-threshold obstruction](https://closedform.nishachay.tech/e/openai-math-2026-240/) - Entry 240 · Claims Shelah's eventual categoricity conjecture in ZFC, and shows under CH that its proposed exact threshold fails. (Formal proof listed) - [Nonsingular systems of equations over arbitrary groups](https://closedform.nishachay.tech/e/openai-math-2026-256/) - Entry 256 · Claims every nonsingular system of equations over any group is solvable in a larger group, settling Howie's conjecture. (Formal proof listed) - [Failure of Kohn–Sham ensemble representation](https://closedform.nishachay.tech/e/openai-math-2026-278/) - Entry 278 · Claims a three-electron molecule whose ground-state density no Kohn–Sham ensemble can reproduce. (Claimed, not checked) - [The Kadison–Ringrose cohomology conjecture](https://closedform.nishachay.tech/e/openai-math-2026-295/) - Entry 295 · Proves bounded Hochschild cohomology of von Neumann algebras vanishes in degrees two and up, settling Kadison–Ringrose. (Formal proof listed) - [The generator problem for finite factors](https://closedform.nishachay.tech/e/openai-math-2026-296/) - Entry 296 · Every type II₁ factor with separable predual is generated by a single operator. (Formal proof listed) - [A counterexample to Wall's finite D(2) problem](https://closedform.nishachay.tech/e/openai-math-2026-321/) - Entry 321 · A paper claims a negative answer to Wall's D(2) problem: a finite 3-complex that passes the test but has no 2-d model. (Claimed, not checked) - [Independence of the separable quotient problem](https://closedform.nishachay.tech/e/openai-math-2026-323/) - Entry 323 · The separable quotient problem cannot be settled from the usual axioms, a paper claims; Lean checks the CH half. (Formal proof listed) - [Nonexpansive fixed points in reflexive Banach spaces](https://closedform.nishachay.tech/e/openai-math-2026-328/) - Entry 328 · Proves that every nonexpansive map of a closed bounded convex set in a reflexive Banach space has a fixed point. (Formal proof listed) - [Negative Kähler curvature without bounded holomorphic coordinates](https://closedform.nishachay.tech/e/openai-math-2026-359/) - Entry 359 · Proves a complete Kähler threefold with pinched negative curvature can lack bounded holomorphic coordinates. (Formal proof listed) - [Interior C^(1,α) regularity for infinity-harmonic functions](https://closedform.nishachay.tech/e/openai-math-2026-377/) - Entry 377 · Claims the gradient of every infinity-harmonic function is Hölder continuous inside its domain, in dimension 3 and up. 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