OpenAI solved an 80-year-old puzzle

An internal OpenAI model has resolved an 80-year-old problem in discrete geometry by disproving a conjecture posed by the late Hungarian mathematician Paul Erdős in 1946. The unit-distance problem asked for the maximum number of pairs of points on a plane that can be exactly one unit apart, with Erdős conjecturing the number grew only slightly faster than the number of dots (linearly). OpenAI’s model discovered an infinite family of configurations that achieve a polynomial improvement, far exceeding the linear bound mathematicians had long assumed was optimal. The company announced the discovery alongside 19 pages of endorsements from independent mathematicians, including Daniel Litt of the University of Toronto and Timothy Gowers of the Collège de France. Gowers stated the proof would have been accepted without hesitation if submitted to the *Annals of Mathematics*, calling it a landmark in AI-driven mathematics. The result was produced by a general-purpose reasoning model, not one specifically trained for mathematics, highlighting its broad applicability. Erdős had offered monetary rewards—initially $300, later increased to $500—for a solution to his conjecture. His work spanned number theory, combinatorics, and graph theory, with over 1,500 published papers. The breakthrough underscores AI’s ability to synthesize diverse mathematical disciplines and persist in problem-solving, unlike human mathematicians who often focus on narrower expertise. OpenAI emphasized the discovery as part of broader efforts to test AI’s contributions to cutting-edge research. The model was evaluated on a collection of Erdős problems, and this case resolved an open question autonomously. The achievement reflects the growing competition among AI firms like OpenAI and Anthropic to demonstrate advanced capabilities, with financial and research implications. The unit-distance problem had long been a favorite of Erdős, who dedicated nearly all his time to mathematics. His conjecture shaped decades of research, and its disproof marks a rare instance where AI has autonomously produced a result deemed mathematically rigorous by experts.