OpenAI's reasoning model autonomously solves 80-year-old discrete geometry problem, bridging algebraic number theory and combinatorics
An OpenAI general-purpose reasoning model has disproved a central conjecture in discrete geometry—the planar unit distance problem posed by Paul Erdős in 1946—by discovering an unexpected connection to algebraic number theory. The model constructed an infinite family of point configurations yielding polynomial improvement over the best-known constructions (square grids), a result validated by external mathematicians including Fields medalist Tim Gowers, who called it "a milestone in AI mathematics." Notably, the proof was not generated by a system trained specifically for mathematics or targeted at this problem, but emerged from testing a general-purpose reasoning model on a collection of Erdős problems. The significance extends beyond the specific result. The proof demonstrates that AI can hold together complex multi-step arguments, connect ideas across distant mathematical domains that remain siloed due to specialization, and surface promising paths experts may not have prioritized. Leading number theorist Arul Shankar stated the result shows "current AI models go beyond just helpers to human mathematicians – they are capable of having original ingenious ideas, and then carrying them out to fruition." The companion work by external mathematicians revealed that algebraic number theory has far deeper implications for discrete geometry than previously suspected, suggesting this breakthrough may catalyze further discoveries across mathematics and adjacent fields like biology, physics, and materials science.