OpenAI model disproves Erdős conjecture

1/ OpenAI said on May 20 that one of its internal reasoning models produced a disproof of Paul Erdős’s 1946 unit distance conjecture, a long-standing problem in discrete geometry. The company published a research post, a proof PDF, a companion remarks paper by outside mathematicians, and a rewritten summary of the model’s reasoning. (openai.com) 2/ The problem sounds simple: given \(n\) points in the plane, how many pairs can be exactly distance 1 apart? Erdős showed constructions with slightly more than linear many unit distances and conjectured the true maximum was only \(n^{1+o(1)}\), while the best known upper bound remained \(O(n^{4/3})\). (openai.com) 3/ OpenAI’s claim is not that the model tightened an existing bound. It says the model found an infinite family of point sets with at least \(n^{1+\varepsilon}\) unit-distance pairs for some fixed \(\varepsilon>0\), which is enough to refute Erdős’s conjectured near-linear upper bound. (arxiv.org) 4/ In plain English: the conjecture said the count should grow only barely faster than \(n\). A construction with a genuine polynomial improvement — \(n^{1+\varepsilon}\) for fixed \(\varepsilon\) — breaks that picture. The old upper bound \(O(n^{4/3})\) still stands; what changed is the lower-bound side and the belief about what extremal examples can do. (arxiv.org) 5/ OpenAI said the result came from “a new general-purpose reasoning model,” not a system trained specifically for mathematics, nor one scaffolded to search proof strategies for this problem in particular. The company called it “the first time that a prominent open problem, central to a subfield of mathematics, has been solved autonomously by AI.” (openai.com) 6/ The outside verification matters because OpenAI had already been criticized for overstating earlier Erdős-related claims. TechCrunch reported that a 2025 post by then-OpenAI executive Kevin Weil claiming GPT-5 had solved unsolved Erdős problems was later withdrawn after mathematicians said the purported solutions were already in the literature. (techcrunch.com) 7/ This time, the companion remarks paper is signed by Noga Alon, Thomas Bloom, W. T. Gowers, Daniel Litt, Will Sawin, Arul Shankar, Jacob Tsimerman, Victor Wang and Melanie Matchett Wood. The paper describes itself as a “human-verified version” of the OpenAI-generated counterexample. (arxiv.org) 8/ The strongest outside language in the published materials comes from named mathematicians. Tim Gowers, quoted by OpenAI, called the result “a milestone in AI mathematics.” Arul Shankar said the paper showed current AI models can have “original ingenious ideas” and carry them through. (openai.com) 9/ What did the proof actually use? The remarks paper says the argument brings in algebraic number theory, including number fields, class numbers and prime-splitting behavior, to build planar point sets with unusually many unit distances. The authors say those ideas can be linked, in retrospect, to work by Ellenberg-Venkatesh, Golod-Shafarevich, and Hajir-Maire-Ramakrishna. (arxiv.org) 10/ That is part of why mathematicians found the result notable. OpenAI said the proof imported “unexpected, sophisticated ideas from algebraic number theory” into an elementary-looking geometry question. The remarks paper includes a quoted summary of the model’s reasoning that explicitly turns to number fields as a source of possible counterexamples. (openai.com) 11/ The “diagrammatic reasoning steps and code” point also checks out in substance, though the public materials are split across formats. OpenAI linked a formal proof PDF, a companion remarks paper, and a rewritten chain-of-thought summary; the company’s post says the model was evaluated as part of a broader effort on Erdős problems. (openai.com) 12/ One thing to keep straight: this is a disproof of a conjecture about the asymptotic maximum number of unit distances in the Euclidean plane, not a full solution of every aspect of the unit distance problem. The gap to the best known upper bound remains. (arxiv.org) 13/ The immediate next places to look are the proof itself, the companion remarks, and the rewritten reasoning summary that OpenAI published on May 20. Those documents contain the formal statement, the human-digested exposition, and the model-derived path to the construction. (cdn.openai.com)