SAN FRANCISCO — OpenAI's internal AI model solved the planar unit distance problem, an 80-year-old mathematics question first posed by Hungarian mathematician Paul Erdős in 1946. The AI-generated solution exceeded the previously established human upper bound for the problem, which had stood since 1984.
The planar unit distance problem asks for the maximum number of pairs of points that can exist one unit apart on a two-dimensional plane. Erdős had conjectured that this number would rise slightly faster than the number of points. The AI model, which was not specifically trained for mathematics or this problem, used a novel approach that replaced a longstanding working theory associated with the problem.
OpenAI representatives wrote, "This proof is an important milestone for the math and AI communities. It marks the first time that a prominent open problem, central to a subfield of mathematics, has been solved autonomously by AI." They added, "These ideas were well-known to algebraic number theorists, but it came as a great surprise that these concepts have implications for geometric questions."
External human mathematicians reviewed and confirmed the AI-generated proof and co-authored a companion paper explaining its context and implications. OpenAI stated that the technology is intended to improve mathematicians' work, not replace them, and suggested the result serves as a proof of concept that AI can be applied to frontier research beyond this specific problem.
Thomas Bloom, a mathematician at the University of Manchester, wrote, "While the original proof produced by AI was completely valid, it was improved by the human researchers at OpenAI and the many other mathematicians involved in the present paper." He added, "The human still plays a vital role in discussing, digesting and improving this proof, and exploring its consequences." Tim Gowers, a professor of mathematics at the University of Cambridge, wrote, "There is no doubt that the solution to the unit-distance problem is a milestone in AI mathematics: if a human had written the paper and submitted it to the Annals of Mathematics and I had been asked for a quick opinion, I would have recommended acceptance without any hesitation." He also stated, "No previous AI-generated proof has come close to that."