80-year-old geometry mystery cracked by OpenAI using deep number theory

OpenAI says a reasoning model helped solve the planar unit distance problem, a famous Erdős-era open question in discrete geometry.

The model found an infinite family of point sets with more unit-distance pairs than the long-assumed square-grid pattern.

Mathematicians later refined the proof and tied the geometry problem to deep algebraic number theory, including class field tower ideas.

The result is being seen as a major AI-for-math breakthrough, showing reasoning models may contribute novel research ideas beyond theorem proving.