OpenAI Model Solves 80-Year-Old Geometry Problem

OpenAI has announced that one of its internal models has autonomously solved the planar unit distance problem, a question in discrete geometry that has remained open since 1946. The result was detailed in an announcement on OpenAI's website and verified by a group of external mathematicians.

The problem, originally posed by Paul Erdős, asks how many pairs of points among n points in a plane can be exactly one unit apart. For decades, mathematicians believed that the square grid construction was nearly optimal, achieving a growth rate of n to the power of 1 plus a term approaching zero. The OpenAI model produced a new family of configurations that yield a polynomial improvement, disproving this long-standing conjecture.

The proof, which introduces ideas from algebraic number theory, constructs point arrangements with at least n raised to the power of 1 plus delta unit distance pairs, where delta is a positive constant. A refinement by Princeton professor Will Sawin established delta as 0.014. The proof was confirmed by mathematicians including Noga Alon, Tim Gowers, Arul Shankar, and Jacob Tsimerman.

Unlike systems designed specifically for mathematics, the model used was a general reasoning system evaluated on a collection of Erdős problems. It independently developed the proof without being targeted at the unit distance problem. Mathematicians involved in the verification described the result as a milestone for AI in mathematics, highlighting its unexpected use of algebraic number theory to solve a geometric question.