OpenAI disproves 1946 geometry conjecture

OpenAI said on May 20 that one of its internal reasoning models had disproved a longstanding conjecture in discrete geometry tied to the planar unit distance problem, a question first posed by Paul Erdős in 1946. The company said the model found an infinite family of constructions that beats the square-grid style examples mathematicians had long believed were essentially optimal. (openai.com) OpenAI published a research post, the proof itself and companion remarks from outside mathematicians the same day. TechCrunch reported that some of the mathematicians who had criticized an earlier OpenAI math claim backed this result. ### Which conjecture did OpenAI say it disproved? Paul Erdős asked in 1946 how many pairs of points in the plane can be exactly one unit apart among \(n\) points, a problem now known as the planar unit distance problem. OpenAI said the unresolved conjecture was the field’s long-running belief that square-grid constructions were essentially best possible, or close to it, for maximizing those unit-distance pairs. OpenAI said its model produced an infinite family of counterexamples that improves on that approach by a polynomial factor. In its research post, the company described the result as a disproof of a “central conjecture in discrete geometry,” not a full resolution of the entire unit distance problem. (openai.com) ### Why are mathematicians taking this claim more seriously than the last one? TechCrunch reported that OpenAI’s previous public claim about solving Erdős problems unraveled after the company cited results that were already known in the literature. Thomas Bloom, who runs the Erdős Problems website, had called that earlier episode “a dramatic misrepresentation,” according to TechCrunch. (openai.com) This time, TechCrunch said Bloom, Princeton combinatorialist Noga Alon and Harvard mathematician Melanie Wood were among the named mathematicians supporting the new result in companion remarks published alongside OpenAI’s announcement. (openai.com) OpenAI also said the proof was checked by a group of external mathematicians. ### What did OpenAI say the model actually did? OpenAI said the proof came from a new general-purpose reasoning model rather than a system trained only for mathematics, scaffolded to search proof strategies, or aimed specifically at the unit distance problem. The company said it had been testing advanced models on a collection of Erdős problems as part of a broader effort to see whether they could contribute to frontier research. (techcrunch.com) In the same post, OpenAI said the argument brought ideas from algebraic number theory into an elementary geometric question. The company called that cross-field connection part of what made the result notable. (openai.com) ### What have named experts said about the result? Tim Gowers, a Fields Medal-winning mathematician, said in the companion paper that the result was “a milestone in AI mathematics,” according to OpenAI’s post. Arul Shankar, a number theorist, said the paper showed current AI models could produce “original ingenious ideas” and carry them through. (openai.com) Thomas Bloom also praised the result in comments cited by TechCrunch, saying AI was helping researchers explore “the cathedral of mathematics” in new ways. That support matters because Bloom was among the people who publicly challenged OpenAI’s earlier math announcement. (openai.com) ### What comes next? OpenAI’s next step is already public: the company posted the proof and companion remarks on May 20 for broader mathematical scrutiny. The immediate test will be whether other researchers confirm the argument and build on the new constructions in follow-up work on the unit distance problem. (openai.com) (techcrunch.com)