OpenAI solves planar distance problem

OpenAI said on May 20 that one of its internal general-purpose reasoning models had disproved a longstanding conjecture in discrete geometry by finding a new family of constructions for the planar unit distance problem. The problem, posed by Paul Erdős in 1946, asks how many pairs of points can be exactly one unit apart in a set of \(n\) points in the plane. For decades, mathematicians broadly believed square-grid-type constructions were essentially optimal. OpenAI said its model instead produced an infinite family with a polynomial improvement, and outside mathematicians published a companion note saying they had checked the argument. ### What exactly is the conjecture that OpenAI says it disproved? Paul Erdős conjectured in 1946 that the maximum number of unit-distance pairs, usually written \(\nu(n)\), should stay near \(n^{1+o(1)}\), with a more specific bound of the form \(n^{1+C/\log\log n}\) for some constant \(C\), according to OpenAI’s proof and the companion remarks. The classical lower-bound example came from grid constructions, which already give slightly more than linear growth, but not a fixed power above 1. (openai.com) The new result claims something stronger in the opposite direction: OpenAI’s proof states that there exists a fixed \(\delta>0\) such that \(\nu(n)\ge n^{1+\delta}\) for infinitely many \(n\). That is the statement that contradicts the old conjecture. ### What did the model actually find? OpenAI said the model found “an infinite family of examples” that beats the old square-grid intuition. (cdn.openai.com) The proof document says the construction runs through towers of totally real number fields with 3-power Galois groups, then uses those fields to build high-dimensional lattices with many elements whose complex embeddings all have absolute value 1. The companion remarks by Noga Alon, Thomas Bloom, W. T. Gowers, Daniel Litt, Will Sawin, Arul Shankar, Jacob Tsimerman, Victor Wang and Melanie Matchett Wood describe their paper as a “human-verified” and simplified version of the OpenAI-generated counterexample. They say the argument draws on ideas linked, in retrospect, to prior work by Ellenberg-Venkatesh, Golod-Shafarevich, and Hajir-Maire-Ramakrishna. (openai.com) ### Who checked it? OpenAI said the proof “has been checked by a group of external mathematicians.” In the companion paper, nine mathematicians are listed as authors, including Princeton combinatorialist Noga Alon, Fields Medalist Tim Gowers, and number theorists Arul Shankar and Melanie Matchett Wood. Tim Gowers wrote in the companion paper, as quoted by OpenAI, that the result is “a milestone in AI mathematics.” Arul Shankar said, in OpenAI’s account, that the paper shows current AI models are “capable of having original ingenious ideas, and then carrying them out to fruition.” (cdn.openai.com) ### Why are people treating this as unusual? OpenAI said the proof came from “a new general-purpose reasoning model,” not from a system trained only for mathematics, scaffolded for proof search, or aimed specifically at the unit distance problem. (openai.com) The company said it had been testing whether advanced models could contribute to frontier research by evaluating them on a set of Erdős problems. That makes the claim narrower and more specific than a general statement that AI can now do mathematics on its own. What OpenAI has publicly documented is one result, on one prominent open problem, with a proof and a companion verification note released alongside the announcement. ### What has OpenAI released for others to inspect? OpenAI published three main documents on May 20: a public announcement, a proof titled *Planar Point Sets with Many Unit Distances*, and a rewritten chain-of-thought summary. (openai.com) The outside mathematicians’ companion paper, *Remarks on the Disproof of the Unit Distance Conjecture*, was also posted through OpenAI’s site. I could verify OpenAI’s announcement, the proof, and the external companion remarks. I could not independently verify the “48% success rate” claim from the social-media circulation using primary-source documents available on OpenAI’s site, so I have not included that figure as established fact. (openai.com)