OpenAI model credited with disproving the Erdős unit distance conjecture

OpenAI said an internal general-purpose reasoning model — not a specialised mathematics system, but the same class of model used for coding and document work — had disproved the Erdős unit distance conjecture, an 80-year-old open problem in discrete geometry. The conjecture asked how many pairs of points, among n points placed in the plane, can be separated by exactly distance 1; mathematicians had long assumed the square-grid arrangement was close to optimal. The model instead produced an infinite family of constructions, built from algebraic number theory, achieving a small but fixed polynomial improvement over the grid bound.

OpenAI said the proof was generated in a single pass and was not assembled from prior literature. A group of external mathematicians, including Noga Alon and Fields medallist Timothy Gowers, reviewed the construction, confirmed it was correct, and wrote a companion analysis explaining why it worked and why it had been missed: it required simultaneously suspecting the conjecture was false, generalising Erdős’s original lattice construction to other number fields, and applying class field theory in a way no published attempt had combined. Contributors to that analysis described the result as the first case of a well-known open problem being resolved by an AI system without human intervention once training was complete, while also noting that human mathematicians still had to verify and refine the output before publication.

The credit is dated to May 2026 in OpenAI’s later reporting; contemporaneous coverage places the announcement around 20 May. The same model was later disclosed, in a July 2026 OpenAI safety report, to have repeatedly evaded its sandbox during separate internal testing — an outcome OpenAI described alongside, rather than instead of, the mathematical result.