List of mathematical discoveries by artificial intelligence
From Wikipedia, the free encyclopedia
Beginning in 2026, large language models, including reasoning models, have made increasing progress in generating mathematical proofs at research level.[1][2][3] Most such proofs have been conducted using models from OpenAI and Anthropic.
The degree of human intervention in AI-generated mathematical proofs may vary and remain unclear. Subjects responsible for publishing such proofs, including individuals and AI companies such as OpenAI, often fail to disclose used prompts, structure of pipelines used as a model workflow, or specific models used for generating proofs. The publication of such proofs may not satisfy standards of science articles. It may not have passed independent review or peer review, and instead be published on corporate websites, personal websites, or social media.
List
- In May 2026, OpenAI announced that an internal model had produced a disproof of the unit distance problem.[4][5]
- In July 2026, mathematician Levent Alpöge presented an explicit counterexample to the Jacobian conjecture in dimension 3, which he found with Claude Fable 5.[6][7]
- In August 2026, OpenAI announced ten results achieved by an internal version of Astra, an unreleased model,[8] including a construction establishing the existence of non-sofic groups. However, a supposed disproof of Connes's rigidity conjecture was later refuted.[9] Neither the original proofs nor the refutation have yet undergone peer review.
- In August 2026, Anthropic announced that an unreleased version of Claude increased the proven share of Riemann zeta function zeros on the critical line from 41.6% to 67.2%.[10][11][12]
- In August 2026, Levent Alpöge and Ava Howell used Claude to find the two elliptic curves with the largest known rank (at least 30 and 31, respectively).[13]
- In August 2026, Alpöge claimed to have used Claude to prove that the 6-sphere admits a complex structure. The result has not yet been independently verified. If confirmed, this would resolve the 78-year old Hopf problem.[14]
- In August 2026, an anonymous human under the pseudonym DottedCalculator (claiming authorship only by GPT-5.6 Sol) uploaded a paper to Github, positing an improvement on the Erdős–Rankin conjecture (Erdős Problem 4). The result has been fully formalized in Lean. According to the paper, there are infinitely many gaps between primes such that for some constant . This improves on the original proof of the conjecture by Ford, Green, Konyagin, Maynard and Tao from 2014. According to Green, the new result uses a new but relatively elementary approach that would have been strong enough to independently solve the original problem without their 2014 paper, but which could have been discovered in the 1960s.[15][16]
- In August 2026, OpenAI announced progress on the twin prime conjecture. They claimed to have proved that there are infinitely many prime numbers that are at most 186 apart, an improvement on the previous bound of 246 that had been established in 2014 by the Polymath Project. The OpenAI result has not yet been independently verified.[17]
- In September 2026, Anthropic published a claimed proof of the dying percolation conjecture, one of the central open problems in percolation theory, together with its formalization in Lean.[18] The result was obtained by proving an equivalent conjecture by Gady Kozma and Shaha Nitzan. Building on the result, Ahmed Bou-Rabee from the University of Pennsylvania published a claimed proof of the remaining Kozma-Nitzan conjectures, which he found the following day with ChatGPT Sol 5.1 Ultra and formalized in Lean with Claude Fable 5.1.[19]
- In September 2026, Anthropic published the first complete computer-checked proof of Fermat's Last Theorem, found with the help of Claude over the course of 11 days. It formalizes a simplified version of the original proof found by Andrew Wiles in 1994. Work on formalizing the theorem had started in the 2000s. Anthropic's proof in Lean is 13 million lines long, the largest Lean proof written to date.[20][21][22]
See also
- Computer-assisted proof
- List of conjectures by Paul Erdős, several of which have recently been solved with AI assistance
- Leiden Declaration on Artificial Intelligence and Mathematics