AI Solves 80-Year-Old Mathematical Problem on Point Distances

An artificial intelligence model has successfully solved a mathematical problem that has remained open for 80 years, marking a significant milestone in the field of AI mathematics.

The problem, first posed by Hungarian mathematician Paul Erdős in 1946, concerns the maximum number of pairs of points separated by a unit distance on a two-dimensional plane. Erdős proposed that this maximum number grows slightly faster than the number of points themselves.

Until recently, the most precise human-derived upper bound for this problem was established in 1984. However, OpenAI revealed in a blog post that one of its internal AI models has found arrangements surpassing Erdős's bound.

Importantly, OpenAI stated that the general reasoning model used to solve the problem was not specifically trained for this challenge or even for mathematics in general, according to Live Science, a specialized science and technology news outlet.

Representatives of OpenAI wrote in the post: "This proof represents a significant milestone for both the mathematics and AI communities. It is the first time an AI has independently solved a prominent open problem central to a branch of mathematics."

OpenAI scientists explained that their model employed an entirely new approach to replace a working theory commonly used in the unit distance problem on the plane.

They added: "These ideas were well known among algebraic number theory experts, but it was quite surprising that these concepts would have implications for geometric problems."

OpenAI announced that this outcome is the first instance of AI independently solving an open problem in any field.

Nevertheless, acknowledging previous public criticism of claims that AI would replace humans, the company emphasized that the technology aims to assist mathematicians rather than substitute them. External mathematicians were invited to review and validate the results and co-authored a research paper explaining the context in which the AI reached its conclusions.

Thomas Bloom, a mathematician at the University of Manchester and the curator of the Erdős problems website, wrote in the accompanying paper: "Although the original proof produced by the AI was entirely correct, it was significantly improved thanks to human researchers at OpenAI and many other mathematicians involved in this paper. Humans still play a vital role in discussing, understanding, refining the proof, and exploring its consequences."

Reactions from the mathematics community were highly positive. Tim Gowers, a mathematics professor at the University of Cambridge, stated in a related research paper: "There is no doubt that solving the unit distance problem is a remarkable achievement in AI-based mathematics; if a human had written this paper and submitted it to the Annals of Mathematics, and I were asked for a quick opinion, I would recommend acceptance without hesitation. No previous AI-generated proof has come close to this level."

OpenAI's blog post noted that the result extends beyond the planar unit distance problem, serving as a proof of concept demonstrating AI's potential for broader application in "pioneering research."

It remains to be seen whether this potential will be realized. In October of the previous year, OpenAI representatives, including CEO Kevin Weil and executive Sebastian Bubeck, claimed that GPT-5 had solved 10 Erdős problems never before resolved and made progress on 11 others.

Bubeck later retracted this statement and deleted his initial post after experts, including Bloom, pointed out that those problems had already been solved by mathematicians.