For eighty years, mathematicians believed they understood the answer to a famous puzzle. If you place dots on paper and want as many pairs as possible to sit exactly one unit apart, arranging them in a square grid seemed to be the best solution. Paul Erdős, a Hungarian mathematician, posed this problem in 1946 and believed no better arrangement existed. Most mathematicians agreed with him.
They were wrong. On May 20, 2026, OpenAI announced that one of its general-purpose reasoning models had solved the problem. The AI did not just improve the old answer. It created an entirely new family of point arrangements that contain more unit-distance pairs than any square grid could.
The problem is simple to understand. Place n points anywhere on a flat plane. How many pairs can be exactly one unit apart? Erdős proved that a skewed grid produces pairs that grow slightly faster than n, and he conjectured nothing significantly better was possible.
The AI's solution used unusual tools from pure mathematics. It applied generalizations of Gaussian integers and the Golod-Shafarevich theorem, objects normally studied independently from geometry.
OpenAI sent the proof to leading mathematicians before announcing it publicly. Nine mathematicians verified the solution and published a shorter, human-checked version on arXiv the same day. Within hours, mathematician Will Sawin improved the result further, determining the exact mathematical limits of the method. Erdős offered a prize for anyone who could disprove his conjecture. He never imagined it would be an artificial intelligence that would find the answer.