An OpenAI model refutes a conjecture in discrete geometry that has remained open since 1946

An internal OpenAI model refuted Paul Erdős's 1946 unit distance conjecture, a math breakthrough verified by Tim Gowers and refined by Will Sawin.

An internal OpenAI model has invalidated a nearly eighty-year-old belief regarding the unit distance problem in the plane, posed by Paul Erdős in 1946. The question, simple to state, asks how many pairs of points can be separated by a distance exactly equal to one, among a given number of points placed in the plane. Since its inception, mathematicians believed that square grid constructions approached the optimum. The model produced an infinite family of configurations that perform better, providing a measurable improvement and contradicting the accepted conjecture.

The proof was verified by a group of external mathematicians, who authored an accompanying text detailing the argument and its scope. Its uniqueness lies as much in the result as in the method: it mobilizes tools from algebraic number theory, a seemingly distant field, to address an elementary geometric question. Fields medalist Tim Gowers sees it as a milestone for AI-assisted mathematics, while a subsequent refinement by mathematician Will Sawin, at Princeton, made it possible to clarify the gain obtained.

The notable point lies in the origin of the proof. It comes from a general-purpose reasoning model, and not from a system specifically designed for mathematics nor oriented towards this specific problem. OpenAI sees it as the first autonomous resolution by an AI of a major open problem located at the heart of a subfield, and an indication of what these reasoning capabilities could bring to biology, physics, or medicine.