OpenAI Model Disproves 80-Year-Old Conjecture

The existing conjecture in the planar unit distance problem in discrete geometry, known as Erdős problem 90, has been disproved.

The problem concerns the maximum number of pairs of points at distance 1 among n points in the plane. OpenAI stated that an internal general-purpose reasoning model produced a proof overturning the conjecture raised by Erdős in 1946, and that external mathematicians verified it. Previously, it was widely believed that square grid-type constructions were essentially optimal.

• The result presents examples at the level of n^(1+δ) for infinitely many n.
• OpenAI did not specify δ in the initial proof, but explained that a follow-up theorem by Will Sawin could strengthen it to δ = 0.014.
• The key idea was not geometry but algebraic number theory, using infinite class field towers and Golod–Shafarevich theory.

OpenAI emphasized this as a case where AI autonomously solved a long-unsolved problem using a general reasoning model, rather than a system dedicated to a specific mathematical problem.