OpenAI model disproves Erdős planar unit distance conjecture

An internal reasoning model developed by OpenAI has disproved a longstanding mathematical conjecture about the maximum number of unit distances possible between points on a flat plane, a problem first posed by the legendary mathematician Paul Erdős in 1946. The result, verified by a team of external mathematicians, marks a significant milestone in the application of artificial intelligence to fundamental research.

The planar unit distance problem asks a deceptively simple question: given a set of n points, what is the greatest possible number of pairs that are exactly one unit apart? For decades, the prevailing belief, supported by Erdős's own conjecture, was that arrangements based on a square grid were nearly optimal. These constructions suggested the number of unit-distance pairs could only grow slightly faster than the number of points itself.

OpenAI's model discovered an infinite family of point configurations that yield a polynomial improvement over the old bound, demonstrating that significantly more unit-distance pairs are possible. A refinement by Princeton mathematician Will Sawin established a concrete exponent, showing the new construction achieves at least n^1.014...