OpenAI says model solved Erdős problem

OpenAI said on May 20 that one of its internal reasoning models produced an original proof disproving a longstanding conjecture in discrete geometry first posed by Paul Erdős in 1946. The problem, known as the planar unit distance problem, asks how many pairs of points in the plane can be exactly one unit apart among \(n\) points. OpenAI said mathematicians had long believed square-grid-style constructions were essentially optimal, and that its model found an infinite family of configurations that does better. The company published the proof and a companion set of remarks from outside mathematicians on its website May 20. ### What exactly is the Erdős problem OpenAI says it solved? Paul Erdős posed the planar unit distance problem in 1946, according to OpenAI’s research note. In plain terms, mathematicians ask: given \(n\) points in the plane, how many pairs can be exactly distance 1 apart? OpenAI said the prevailing conjecture held that square-grid constructions were essentially best possible, and that its model produced a disproof by constructing an infinite family with a polynomial improvement. (openai.com) The OpenAI note said the proof brings ideas from algebraic number theory to an elementary geometric question. WinBuzzer reported that, if the argument holds, the construction gives \(n^{1+\delta}\) unit-distance pairs for infinitely many values of \(n\), with Princeton mathematician Will Sawin providing a refinement that puts \(\delta\) at 0.014. ### Why are mathematicians taking this claim more seriously than OpenAI’s last one? (openai.com) TechCrunch reported on May 20 that OpenAI’s previous high-profile Erdős-related claim in 2025 unraveled after the company said GPT-5 had found solutions to unsolved problems that were already in the literature. TechCrunch said Thomas Bloom, who maintains the Erdős Problems website and had called that earlier episode “a dramatic misrepresentation,” is among the mathematicians now backing the new result. (openai.com) OpenAI said the proof “has been checked by a group of external mathematicians,” and published companion remarks from named researchers including Noga Alon, Tim Gowers and Arul Shankar. Gowers, in the companion remarks cited by OpenAI, called the result “a milestone in AI mathematics,” while Shankar said the paper showed current models could have “original ingenious ideas” and carry them through. (techcrunch.com) ### What does OpenAI say the model actually did? OpenAI said the proof came from a “general-purpose reasoning model,” not from a system trained specifically for mathematics or scaffolded to search proof strategies for this problem. The company framed the result as part of a broader effort to test whether advanced models can contribute to frontier research on Erdős problems. (openai.com) TechCrunch reported that OpenAI is presenting the result as the first time AI has autonomously solved a prominent open problem central to a field of mathematics. That characterization remains OpenAI’s claim, but the company’s inclusion of outside mathematicians from the start is a central part of how it is trying to establish credibility this time. (openai.com) ### What is still under review? WinBuzzer reported on May 21 that broader outside review is still focused on technical details, including the 0.014 refinement associated with Sawin. The same report said that figure gives mathematicians a concrete quantity to inspect as they test the larger proof. OpenAI’s own post said the proof has been checked by external mathematicians, but that does not end the normal process of scrutiny. (techcrunch.com) The next step is continued review of the published proof and companion remarks by the wider mathematics community, including researchers such as Bloom, Alon, Gowers, Shankar and Sawin who are already named in the public discussion. (openai.com) (winbuzzer.com)