OpenAI announced on May 21, 2026, that one of its unreleased reasoning models has solved a geometry problem first posed by Hungarian mathematician Paul Erd?s in 1946, according to a company blog post and statements from researchers who reviewed the work. The model produced an original mathematical proof that disproves a long-standing conjecture about the maximum number of equidistant point pairs on a plane, a problem known as the planar unit distance problem [1]. University of Toronto mathematician Arul Shankar said in a statement provided by OpenAI that the model demonstrated “original, ingenious ideas” and was capable of carrying them out to fruition [1]. The findings were published by OpenAI on Wednesday, the company said [2].
Background on the Planar Unit Distance Problem
The planar unit distance problem asks how many pairs of points on a flat surface can be exactly one unit apart. Erd?s posed the question in 1946, and the prevailing theory held that a square grid layout would maximize such pairs, though that conjecture was never proven. Erd?s himself estimated that the number of unit-distance pairs could increase only slightly faster than the number of points as more points are added, according to mathematical literature cited in OpenAI’s announcement [3]. The problem had resisted solution for eight decades, with mathematicians attempting various approaches without success.
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