OpenAI's reasoning model disproves the unit distance conjecture, a problem unsolved for eight decades, by applying algebraic number theory tools to discrete geometry.

OpenAI's internal reasoning model found a new point arrangement that produces roughly one percent more unit-distance pairs per doubling of the point count than the classic square grid, disproving Hungarian mathematician Paul Erdős's 1946 conjecture. Nine external mathematicians verified and commented on the proof in a companion paper.

The model's approach differed from typical human attempts: instead of inflating a single number system progressively (which leads back to the old Erdős bound), it kept the scale fixed within each number system but switched to progressively richer number systems at every step, applying tools from algebraic number theory and class field theory that mathematicians had considered far-fetched for this geometric problem.

Fields Medalist Tim Gowers stated he would have recommended the work for acceptance to the Annals of Mathematics "without any hesitation," calling it "a milestone in AI mathematics." OpenAI describes it as "the first time that a prominent open problem, central to a subfield of mathematics, has been solved autonomously by AI."