On May 21, OpenAI revealed that one of its internal general-purpose reasoning models—still unreleased to the public—managed to independently disprove the Unit Distance Conjecture, proposed in 1946 by legendary Hungarian mathematician Paul Erdős. This conjecture is one of the most famous unsolved problems in combinatorial geometry over the past 80 years. The problem can be stated simply: given any set of n points placed randomly on a plane, what is the maximum possible number of pairs of points whose distance equals exactly 1? Erdős conjectured that this number does not exceed n^(1+o(1)); configurations based on square grids closely approach this upper bound. Without any specialized training, without using theorem-proving tools like Lean, and without relying on scaffolding systems, OpenAI’s model identified infinitely many new configurations that surpass this upper bound in a polynomial sense—thereby providing a counterexample and effectively disproving the conjecture.
What surprised the mathematical community most was the core components of the proof: they originated from algebraic number theory, a branch of mathematics entirely distinct from combinatorial geometry, which deals with factorization properties within field extensions of integers. An independent team of external mathematicians verified the proof and subsequently authored a companion paper; Princeton mathematician Will Sawin later refined the results into an explicit form involving fixed positive exponents. According to OpenAI researcher Noam Brown, the model in question is a general-purpose large language model trained solely for language tasks rather than mathematics; no human intervention was involved during the entire process. The summarized reasoning chain spanned 125 pages; the pivotal breakthrough occurred on page 39. Describing this discovery as “frightening,” Brown noted that the full, unsummarized reasoning chain remains unpublished. OpenAI plans to release this model publicly soon. Commenting on the achievement, Sam Altman remarked: “A general-purpose model has solved an important open problem in mathematics… We’ll likely be saying this kind of thing frequently over the next few years, though today I’m feeling somewhat conflicted.” } OpenAI