OpenAI solves 80-year-old math problem

OpenAI said on May 20 that one of its internal reasoning models had disproved the Erdős unit distance conjecture, a central open problem in discrete geometry first posed in 1946 by Paul Erdős. The company published a research note, a formal proof and a companion remarks paper describing the result and how it was checked. OpenAI said the proof was reviewed by external mathematicians, who also wrote a separate document explaining the argument in a shorter, human-digested form. The claim, if it holds up under wider scrutiny, would place an AI system behind a result on a long-standing research question rather than a contest-style problem. ### Which math problem does OpenAI say it solved? The problem is the planar unit distance problem, which asks how many pairs of points in the plane can be exactly one unit apart among \(n\) points. OpenAI said the question was first posed by Erdős in 1946 and has remained one of the best-known problems in combinatorial geometry. The longstanding conjecture at issue was that the best possible constructions were essentially no better than \(n^{1+o(1)}\) unit-distance pairs. (openai.com) The companion remarks paper says Erdős conjectured an upper bound of \(n^{1+o(1)}\), while the best proved upper bound remains \(O(n^{4/3})\), due to Spencer, Szemerédi and Trotter. ### What exactly did OpenAI claim the model proved? (openai.com) OpenAI said its model produced an infinite family of examples that gives a polynomial improvement over the conjectured bound. In the companion remarks paper, the theorem is stated as the existence of some \(\varepsilon > 0\) and a sequence of point sets \(P_i \subset \mathbb{R}^2\) with \(|P_i| \to \infty\) such that the number of unit distances in \(P_i\) is at least \(|P_i|^{1+\varepsilon}\). (cdn.openai.com) That matters because the result is framed as a counterexample, not a tightening of an existing estimate. OpenAI described it as disproving the prevailing belief that square-grid-style constructions were essentially optimal. ### How does OpenAI say the proof was found? OpenAI said the proof came from a “general-purpose reasoning model,” not from a system trained only for mathematics or one targeted specifically at the unit distance problem. (openai.com) The company said it had evaluated the model on a collection of Erdős problems as part of a broader effort to test whether advanced models can contribute to frontier research. The companion remarks paper says the AI proof drew on ideas connected to algebraic number theory. It quotes part of the model’s reasoning as focusing on algebraic realizations of extremal examples and concluding that “Number fields deserve a closer look,” though the paper adds that related ideas were not wholly unprecedented among experts. ### Who checked the result? (openai.com) OpenAI said a group of external mathematicians checked the proof and wrote a companion paper with background and commentary. That paper is signed by Noga Alon, Thomas F. Bloom, W. T. Gowers, Daniel Litt, Will Sawin, Arul Shankar, Jacob Tsimerman, Victor Wang and Melanie Matchett Wood. OpenAI also highlighted comments from named mathematicians in its post. (cdn.openai.com) It said Fields Medalist Tim Gowers called the result “a milestone in AI mathematics,” and quoted Arul Shankar as saying current AI models are “capable of having original ingenious ideas, and then carrying them out to fruition.” ### What should readers watch next? The next step is ordinary mathematical verification. (openai.com) OpenAI has posted the proof and the companion remarks publicly, and the remarks paper describes itself as a human-verified version of the OpenAI-generated counterexample. Wider peer review will now depend on other mathematicians examining the posted argument and any supporting materials in the days and weeks ahead.