# Open problems from the 1940s > 12 AI-made results on questions first raised the 1940s, each explained in plain English. Results on questions first raised the 1940s, going by the… Canonical: https://closedform.nishachay.tech/open-since/1940s/ Last updated: 2026-10-14 Results on questions first raised the 1940s, going by the earliest step each entry records. 12 AI-made results on questions first raised the 1940s, from 16 papers in the OpenAI math collection, each explained in plain English. 7 have a formal Lean proof listed; 5 are claimed in a paper and not yet machine-checked. They span 6 fields, led by combinatorics (3), dynamical systems and ergodic theory (3) and algebra (2). The oldest question here dates back to 1940: [A counterexample to Kaplansky’s zero-divisor conjecture](https://closedform.nishachay.tech/e/openai-math-2026-196/). Results rated Landmark include [Hilbert’s tenth problem over ℚ](https://closedform.nishachay.tech/e/openai-math-2026-004/) and [Rokhlin’s multiple-mixing problem](https://closedform.nishachay.tech/e/openai-math-2026-145/). ## Results (12) - [Hilbert’s tenth problem over ℚ](https://closedform.nishachay.tech/e/openai-math-2026-004/) - Entry 004 · No algorithm can decide whether a whole-number polynomial equation has a solution in fractions, the paper claims. (Claimed, not checked) - [The weak inhomogeneous Duffin–Schaeffer conjecture](https://closedform.nishachay.tech/e/openai-math-2026-022/) - Entry 022 · For any fixed shift, a weighted sum decides whether almost every number is well approximated. (Claimed, not checked) - [Banach’s simple Lebesgue-spectrum problem](https://closedform.nishachay.tech/e/openai-math-2026-144/) - Entry 144 · A smooth map of a 3D torus where one measurement’s past and future build every average-zero measurement. (Formal proof listed) - [Rokhlin’s multiple-mixing problem](https://closedform.nishachay.tech/e/openai-math-2026-145/) - Entry 145 · If long waits make any two moments independent, they make every finite group of moments independent. (Formal proof listed) - [The entropy-rate dimension formula for self-similar measures](https://closedform.nishachay.tech/e/openai-math-2026-148/) - Entry 148 · A single formula gives the dimension of every self-similar measure on the line, even when its pieces collide. (Formal proof listed) - [Graph coloring, clique minors, and Colin de Verdière invariants](https://closedform.nishachay.tech/e/openai-math-2026-157/) - Entry 157 · Proves Hadwiger's 1943 coloring conjecture is false, and that a fixed multiple of the minor size still bounds coloring. (Formal proof listed) - [The higher-dimensional Erdős distinct-distances conjecture](https://closedform.nishachay.tech/e/openai-math-2026-166/) - Entry 166 · Any n points in d ≥ 3 dimensions span at least a constant times n^(2/d) distinct distances, the paper claims. (Claimed, not checked) - [Planar distinct distances and unit-distance bounds](https://closedform.nishachay.tech/e/openai-math-2026-167/) - Entry 167 · Proves almost every point in a large planar set sees nearly n distances, and lowers the unit-distance exponent 4/3. (Formal proof listed) - [A counterexample to Kaplansky’s zero-divisor conjecture](https://closedform.nishachay.tech/e/openai-math-2026-196/) - Entry 196 · A group with no looping moves still lets two nonzero sums multiply to zero. (Formal proof listed) - [A counterexample to Kurosh’s division-ring problem](https://closedform.nishachay.tech/e/openai-math-2026-201/) - Entry 201 · A division ring made from two elements, each element algebraic, can still be infinite-dimensional over its center. (Claimed, not checked) - [The joint critical Ashkin–Teller current limit](https://closedform.nishachay.tech/e/openai-math-2026-233/) - Entry 233 · Heights and current clusters of the critical Ashkin–Teller model converge together to a free field and its local sets. (Claimed, not checked) - [Isomorphism of the free group factors](https://closedform.nishachay.tech/e/openai-math-2026-287/) - Entry 287 · Proves, with a Lean check, that the free group factors L(F₂) and L(F₃) are isomorphic, settling a decades-old problem. 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