# Open problems from before 1940 > 16 AI-made results on questions first raised before 1940, each explained in plain English. Results on questions first raised before 1940, going by the… Canonical: https://closedform.nishachay.tech/open-since/before-1940/ Last updated: 2026-10-14 Results on questions first raised before 1940, going by the earliest step each entry records. 16 AI-made results on questions first raised before 1940, from 33 papers in the OpenAI math collection, each explained in plain English. 9 have a formal Lean proof listed; 7 are claimed in a paper and not yet machine-checked. They span 9 fields, led by differential geometry (3), combinatorics (2) and dynamical systems and ergodic theory (2). The oldest question here dates back to 1896: [The quasi-Riemann hypothesis](https://closedform.nishachay.tech/e/openai-math-2026-003/). Results rated Landmark include [Primitive roots for every admissible integer base](https://closedform.nishachay.tech/e/openai-math-2026-029/) and [Koebe’s circle-domain conjecture](https://closedform.nishachay.tech/e/openai-math-2026-071/). ## Results (16) - [The quasi-Riemann hypothesis](https://closedform.nishachay.tech/e/openai-math-2026-003/) - Entry 003 · No zero of Riemann's zeta function, or of any Dirichlet L-function, lies to the right of real part 7/8. (Formal proof listed) - [Primitive roots for every admissible integer base](https://closedform.nishachay.tech/e/openai-math-2026-029/) - Entry 029 · Every whole number except −1 and perfect squares is claimed to be a primitive root for infinitely many primes. (Claimed, not checked) - [Koebe’s circle-domain conjecture](https://closedform.nishachay.tech/e/openai-math-2026-071/) - Entry 071 · Every region of the sphere, however many holes it has, can be reshaped so that each hole is round. (Formal proof listed) - [The three-dimensional Bochner–Riesz conjecture](https://closedform.nishachay.tech/e/openai-math-2026-078/) - Entry 078 · A claimed proof of the Bochner–Riesz conjecture in three dimensions, in its strict-order form. (Claimed, not checked) - [The Mahler conjectures, functional inequalities and polar-product symplectic width](https://closedform.nishachay.tech/e/openai-math-2026-087/) - Entry 087 · Symmetric convex bodies cannot beat Mahler's volume bound. (Formal proof listed) - [The near-boundary Birkhoff conjecture](https://closedform.nishachay.tech/e/openai-math-2026-147/) - Entry 147 · Two papers claim that a smooth billiard table with an unbroken band of caustics along its edge must be an ellipse. (Claimed, not checked) - [Arithmetic classification and non-Pisot singularity for Bernoulli convolutions](https://closedform.nishachay.tech/e/openai-math-2026-153/) - Entry 153 · Claimed singular Bernoulli convolutions beyond Pisot reciprocals, plus an arithmetic test for every parameter. (Claimed, not checked) - [Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi bounds](https://closedform.nishachay.tech/e/openai-math-2026-159/) - Entry 159 · Any set of whole numbers whose reciprocals add to infinity holds evenly spaced runs of every length. (Formal proof listed) - [Sharp logarithmic exponents for off-diagonal Ramsey numbers](https://closedform.nishachay.tech/e/openai-math-2026-170/) - Entry 170 · For every fixed s ≥ 5, the Ramsey number r(s,t) now has its exact power of log t, up to a vanishing error. (Formal proof listed) - [An infinite finitely presented residually finite 2-group and a finitely presented nil algebra](https://closedform.nishachay.tech/e/openai-math-2026-247/) - Entry 247 · Claims an infinite finitely presented group where every element has finite order, answering a Burnside question. (Formal proof listed) - [Positive-temperature Bose–Einstein condensation and exact quantum depletion](https://closedform.nishachay.tech/e/openai-math-2026-267/) - Entry 267 · Proves condensation in every ground state of the dilute hard-sphere Bose gas; claims thermal and depletion laws. (Formal proof listed) - [Bloch's law, its lattice correction, and the spherical magnetization law](https://closedform.nishachay.tech/e/openai-math-2026-271/) - Entry 271 · Proves 3D quantum Heisenberg ferromagnets magnetize at low temperature; claims Bloch's T^(3/2) law and its correction. (Formal proof listed) - [The Hilbert–Smith conjecture in every dimension](https://closedform.nishachay.tech/e/openai-math-2026-304/) - Entry 304 · Claims a proof of Hilbert–Smith: a locally compact group acting faithfully on a connected manifold is a Lie group. (Claimed, not checked) - [Sharp Cartan–Hadamard isoperimetry and rigidity](https://closedform.nishachay.tech/e/openai-math-2026-337/) - Entry 337 · Proves sharp Euclidean fillings in CAT(0) spaces and claims the generalized Cartan–Hadamard isoperimetric inequality. (Formal proof listed) - [The metric Blaschke conjecture](https://closedform.nishachay.tech/e/openai-math-2026-344/) - Entry 344 · Claims the metric Blaschke conjecture: injectivity radius equal to diameter forces a round sphere or projective space. (Claimed, not checked) - [Infinitely many closed geodesics on Riemannian spheres and closed three-manifolds](https://closedform.nishachay.tech/e/openai-math-2026-345/) - Entry 345 · Claims every Riemannian metric on a sphere of any dimension has infinitely many geometrically distinct closed geodesics. 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