OpenAI Model Solves 80-Year-Old Erdős Geometry Conjecture

An artificial intelligence model developed by OpenAI has produced a solution to the planar unit distance problem, a conjecture first posed by mathematician Paul Erdős in the twentieth century. The problem asks how many equal-length lines can connect a chosen pattern of dots on an infinite plane.

Erdős had conjectured that grid arrangements would produce the maximum number of such lines, and that the total would be only slightly higher than the number of points.

The OpenAI model used techniques from algebraic number theory to construct lattices in higher dimensions before projecting them into two dimensions. This approach identified point arrangements that yield more connections than the grid patterns Erdős had considered.

Researchers who reviewed the work said the solution demonstrates that the original conjecture was incorrect. The model is described as general-purpose and was not trained specifically for mathematical research.

Rudnev at the University of Bristol said the result was unexpected in his lifetime. Tim Gowers at the University of Cambridge called the solution a milestone in AI mathematics. Will Sawin at Princeton University initially expressed disbelief but later concluded the proof is valid.

Kevin Buzzard at Imperial College London noted that the counterexample combines existing ideas in a novel way. Samuel Mansfield at the University of Manchester said the problem required knowledge across multiple mathematical fields, an area where AI systems may have an advantage.