OpenAI says AI proves Erdős unit distance problem after 80 years

An OpenAI reasoning model has pushed past a nearly 80-year benchmark in discrete geometry, and the crucial test was not the model’s claim but the human verification that followed. OpenAI said the system produced an infinite family of examples that improves, by a polynomial margin, on the square-grid constructions that mathematicians had long treated as essentially optimal for the planar unit distance problem.

The question, first posed by Paul Erdős in 1946, asks for the maximum number of pairs of points in the plane that lie exactly one unit apart. For decades, square-grid arrangements were the standard reference point, and the problem became one of the best known in combinatorial geometry. A 2005 book, Research Problems in Discrete Geometry, described it as possibly the best known and simplest to explain problem in the field, while Erdős himself offered a monetary prize for its resolution.

OpenAI said the proof came from a general-purpose reasoning model, not a system built specifically for mathematics or narrowed to this problem. The company said the result was the first time an AI had autonomously solved a prominent open problem central to a subfield of mathematics. It also said the argument drew on algebraic number theory to settle what had looked like a basic geometric question.

Verification came quickly. OpenAI said the proof was checked by a group of external mathematicians, and a companion note on arXiv was presented as a short, human-verified version of the counterexample. That note was signed by nine mathematicians: Noga Alon, Thomas F. Bloom, W. T. Gowers, Daniel Litt, Will Sawin, Arul Shankar, Jacob Tsimerman, Victor Wang, and Melanie Matchett Wood. The remarks paper said the argument relies crucially on ideas attributable, in retrospect, to Ellenberg-Venkatesh, Golod-Shafarevich, and Hajir-Maire-Ramakrishna.

The reaction from mathematicians was swift and mixed with caution. Fields Medalist Tim Gowers called the result “a milestone in AI mathematics.” Arul Shankar said current AI models are capable of original ingenious ideas and carrying them through to fruition. Daniel Litt called it “the first result produced autonomously by an AI that I find interesting in itself.”

The result does not settle the broader question of what kind of mathematics AI can reliably do. It does show that a general-purpose model, under the right conditions, can generate an argument strong enough to survive expert review and enter the literature as a serious counterexample. Will Sawin reportedly used the same line of reasoning to improve the result soon after OpenAI’s announcement, and Google DeepMind said one of its models had resolved nine lesser Erdős open problems around the same time. For mathematicians, the larger significance is not a single solved puzzle, but the possibility that AI is starting to contribute original, checkable ideas to research that had resisted human methods for generations.