An OpenAI model has disproved a central conjecture in discrete geometry

For nearly 80 years, mathematicians have pondered the planar unit distance problem: how many pairs of points can be exactly one unit apart when n points are placed on a plane? This problem, posed by Paul Erdős in 1946, is a cornerstone of combinatorial geometry, known for its simple statement yet profound difficulty.

An internal OpenAI model has now disproved a long-held conjecture regarding this problem. Previously, it was believed that square grid constructions were optimal for maximizing unit-distance pairs. The AI model provided an infinite family of examples that offer a polynomial improvement, a finding verified by external mathematicians.

This breakthrough is significant not only for its mathematical resolution but also for its origin. The proof emerged from a general-purpose reasoning model, rather than one specifically trained for mathematics. This outcome is part of a broader initiative to assess advanced models' contributions to frontier research, with this instance successfully resolving an open Erdős problem.

The achievement is a major milestone for both the mathematics and AI communities. It marks the first time AI has autonomously solved a prominent open problem central to a mathematical subfield, demonstrating the sophisticated reasoning capabilities these systems now possess. The solution also introduces unexpected and advanced concepts from algebraic number theory to address an elementary geometric question.

Leading experts have lauded this development. Fields medalist Tim Gowers described it as "a milestone in AI mathematics," while number theorist Arul Shankar noted that AI models are now capable of "original ingenious ideas" beyond merely assisting human mathematicians. These sentiments underscore the profound impact of this AI-driven discovery.