OpenAI model challenges Erdős conjecture

OpenAI said on May 20 that an internal reasoning model had produced a proof disproving a long-standing conjecture of Paul Erdős about unit distances in the plane. The claim concerns the planar unit-distance problem, posed by Erdős in 1946: given n points in the plane, how many pairs can sit exactly one unit apart? (openai.com) OpenAI said the model found an infinite family of constructions with a polynomial improvement over the square-grid examples that had shaped belief about the problem for decades. (arxiv.org) A companion paper by outside mathematicians said the argument had been checked and rewritten in human-digested form. ### What was Erdős’s conjecture actually saying? Paul Erdős’s 1946 conjecture said, in effect, that the maximum number of unit-distance pairs among n planar points should stay near n^(1+o(1)), with classic grid constructions believed to be essentially optimal. OpenAI’s proof paper states the opposite: for some fixed δ>0, there are infinitely many n for which the maximum number of unit-distance pairs is at least n^(1+δ). The existing upper bound is still much larger. The OpenAI proof paper and the companion remarks both note that the best known upper bound remains O(n^(4/3)), due to Spencer, Szemerédi and Trotter. (openai.com) That means the new work does not close the problem; it knocks out the old conjectured lower scale. ### Why are mathematicians treating this as more than a flashy AI claim? Tim Gowers, Noga Alon, Thomas Bloom, Daniel Litt, Will Sawin, Arul Shankar, Jacob Tsimerman, Victor Wang and Melanie Matchett Wood signed the companion remarks paper posted on arXiv on May 21. Its abstract calls the note a “short, digested, human-verified version” of the OpenAI-generated counterexample. (cdn.openai.com) OpenAI said the proof was checked by a group of external mathematicians, and quoted Gowers calling the result “a milestone in AI mathematics.” The company also quoted Shankar as saying current AI models are “capable of having original ingenious ideas, and then carrying them out to fruition.” (cdn.openai.com) ### What did the model do that was new? The OpenAI proof paper says the construction does not come from a prettier 2D drawing of the old square grid. It passes through towers of totally real number fields, then uses those fields to build high-dimensional lattices with many vectors whose projections into the plane land at unit distance. (arxiv.org) The companion remarks paper says the argument relies on ideas associated with Ellenberg-Venkatesh, Golod-Shafarevich, and Hajir-Maire-Ramakrishna. It also quotes the model’s reconstructed reasoning as recognizing that “number fields deserve a closer look,” suggesting the route came from importing algebraic-number-theory tools into a geometric extremal problem. (openai.com) ### Where did the widely shared n^1.014 figure come from? Will Sawin posted a separate arXiv preprint on May 21 giving an explicit lower bound. Sawin wrote that there are arbitrarily large n-point planar sets with more than n^1.014 unit-distance pairs, improving on the OpenAI team’s non-explicit exponent and “drastically improving” the previous lower bound. (cdn.openai.com) Sawin’s paper says the OpenAI result was proved by “a team at OpenAI, consisting of Lijie Chen using an internal OpenAI model and Mark Sellke and Mehtaab Sawhney verifying correctness.” He added that the simplified human version of the argument yielded an exponent around 6×10^-38, while his own paper made the exponent explicit and larger. (arxiv.org) ### What should readers watch next? May 20 and May 21 are the key dates in the public record so far. OpenAI has posted the original proof, the companion remarks and a rewritten chain-of-thought summary, while arXiv now also carries Sawin’s explicit-bound follow-up. The next test is standard mathematical scrutiny: whether the proof package holds up through broader expert reading and eventual journal submission by the named participants. (arxiv.org) (openai.com)