OpenAI model disproves Erdős conjecture

OpenAI said on May 20 that one of its internal reasoning models had disproved a central conjecture in discrete geometry, producing a proof on the planar unit-distance problem first posed by Paul Erdős in 1946. The company said the model was not trained specifically for mathematics and was not targeted at that problem in particular. (openai.com) OpenAI published the claim in a research post and linked to both the proof and a companion paper by outside mathematicians. The announcement came two days before Reuters reported that OpenAI had expanded its roster of outside law firms as it navigates lawsuits, deals and a possible future initial public offering. ### Which conjecture did OpenAI say the model disproved? Paul Erdős posed the planar unit-distance problem in 1946: given \(n\) points in the plane, how many pairs can be exactly distance 1 apart? OpenAI said the long-prevailing belief was that square-grid-style constructions were essentially optimal for maximizing those unit-distance pairs. The company said its model disproved that conjecture by finding a better infinite family of constructions. OpenAI said the result produced a polynomial improvement over the old construction family. The company described the problem as one of the best-known questions in combinatorial geometry and cited Princeton mathematician Noga Alon calling it “one of Erdős’ favorite problems.” (openai.com) ### What does OpenAI say the model actually did? OpenAI said the proof emerged during evaluations on a collection of Erdős problems. The company said the system was a new general-purpose reasoning model rather than a math-specialized system or one scaffolded to search proof strategies for this question alone. (openai.com) The company said the argument brought ideas from algebraic number theory to an elementary geometric question. TechCrunch reported that OpenAI presented the result as the first time AI had autonomously solved a prominent open problem central to a field of mathematics. (openai.com) ### Who checked the proof? OpenAI said a group of external mathematicians checked the proof and wrote a companion paper explaining the argument, background and significance. TechCrunch reported that the supporting remarks included mathematicians such as Noga Alon, Melanie Wood and Thomas Bloom, who runs the Erdős Problems website. (openai.com) Tim Gowers, a Fields Medalist quoted by OpenAI, called the result “a milestone in AI mathematics.” OpenAI also quoted Arul Shankar, a number theorist, saying the paper showed current AI models were capable of “original ingenious ideas” and carrying them through. (openai.com) ### Why are mathematicians treating this more seriously than earlier OpenAI claims? TechCrunch reported that OpenAI had previously drawn criticism after a 2025 claim that GPT-5 found solutions to unsolved Erdős problems turned out to refer to solutions already in the literature. TechCrunch said Thomas Bloom had called that earlier episode “a dramatic misrepresentation.” In this case, the outlet reported, OpenAI published companion remarks from outside mathematicians backing the new disproof. (openai.com) OpenAI’s own post said the proof had been checked externally and framed the result as an “important milestone” for both math and AI. (openai.com) That language remains OpenAI’s characterization, not an independent journal decision. ### What else was happening around OpenAI this week? Reuters reported on May 21 that OpenAI had expanded its network of outside counsel to more than a dozen major U.S. law firms as it deals with lawsuits and transactions. Reuters, as cited by The Economic Times, said the company, Chief Executive Sam Altman and lawyers from Wachtell Lipton Rosen & Katz and Morrison & Foerster had just won dismissal of Elon Musk’s lawsuit, removing what Reuters described as a potential hurdle to an IPO that sources told Reuters could come. (techcrunch.com) The next public documents to watch are the companion paper by the external mathematicians and any formal publication or peer-review steps that follow OpenAI’s May 20 release. (openai.com) Reuters also reported that Wachtell and Cooley were tied to OpenAI’s IPO preparations, according to The Information, though OpenAI did not comment on its outside-law-firm spending. (economictimes.indiatimes.com)