OpenAI solves 80-year Erdős problem

OpenAI said on May 20 that one of its reasoning models had disproved a long-standing conjecture tied to Paul Erdős’s 1946 planar unit-distance problem, a question in discrete geometry that asks how many pairs of points in the plane can be exactly one unit apart. (openai.com) The company published the proof and a companion set of remarks on its website. Scientific American reported on May 21 that mathematicians viewed the result as unusually strong for an AI-generated proof, while critics including Gary Marcus said the work still needed broad public checking. (scientificamerican.com) ### What problem did OpenAI say the model solved? Paul Erdős posed the planar unit-distance problem in 1946. OpenAI said the problem asks, for \(n\) points in the plane, how many pairs can be placed exactly distance 1 apart, and said its model found a counterexample that disproves a central conjecture about the best-known constructions. (openai.com) OpenAI described the result not as a full closure of every aspect of the broader unit-distance problem, but as the disproof of a major conjecture within that area of discrete geometry. That distinction matters because the company’s claim is narrower than “solving all of unit-distance geometry,” even though the underlying question dates back nearly 80 years. (openai.com) ### What, exactly, did the company publish? OpenAI published a research post dated May 20 titled “An OpenAI model has disproved a central conjecture in discrete geometry.” The company also linked to the proof itself and to companion remarks, making the underlying argument available for outside review rather than describing it only in a press statement. (openai.com) The OpenAI post said the model was a general-purpose reasoning system rather than software built only for one theorem. The company framed the result as a research milestone in AI-driven mathematics. ### Why are mathematicians treating this as different from earlier AI math claims? (openai.com) Scientific American reported on May 21 that the proof would likely be publishable in a top mathematics journal if human researchers had produced it unaided. The magazine said that would make it the first AI proof to meet that bar in the view of the mathematicians it consulted. (openai.com) TechCrunch reported that some of the mathematicians who had challenged an earlier OpenAI math claim were backing this new result. New Scientist separately reported that mathematicians described the advance as a major moment for AI in mathematics. (openai.com) ### Why are people still asking for verification? Gary Marcus wrote in a Substack post published this week that readers should “read the fine print” and check the mathematics behind OpenAI’s headline. His reaction did not reject the result outright, but it did press for independent examination by specialists outside the company. (scientificamerican.com) That caution reflects the normal way important math claims are handled. OpenAI has made the proof public, but acceptance depends on whether outside mathematicians can verify each step and whether the argument stands up under formal review. That is an inference from the publication of the proof and Marcus’s call for checking, not a statement either side has fully resolved. (techcrunch.com) ### What happens next? Scientific American’s May 21 report said the next stage is scrutiny from working mathematicians, who will test whether the proof is correct and whether its methods can be generalized. OpenAI has already put the proof and remarks online, giving researchers a public record to inspect. (garymarcus.substack.com) Gary Marcus’s post and the public availability of OpenAI’s materials mean the immediate next step is external review, not another company announcement. The proof and companion remarks are the documents mathematicians will use to assess the claim. (openai.com)