OpenAI on May 20 said its internal reasoning model had produced a proof that disproves a key conjecture in the planar unit-distance problem, first posed by Paul Erdős in 1946. The company released the proof alongside a companion paper by nine prominent mathematicians who verified, reorganized and extended the AI-generated argument. The result marks a notable milestone in AI-assisted mathematical discovery, but the broader problem—determining the exact maximum number of unit-distance pairs among n points—remains unsolved.
The result reverses a near-linear upper bound that Erdős had conjectured and that had stood for 80 years. By finding an infinite family of point sets with at least n^(1+δ) unit-distance pairs for some fixed δ>0, the AI-generated counterexample shows the true growth rate is faster than Erdős believed. The verification, published on arXiv hours after OpenAI’s announcement, gives the claim credibility that OpenAI lacked after a 2025 episode in which the company overstated another AI’s achievements on Erdős problems.
The planar unit-distance problem asks: for n points placed anywhere in the plane, what is the maximum number of pairs that are exactly 1 unit apart? Erdős had conjectured that the maximum, denoted ν(n), grows no faster than n^(1+o(1)), roughly linearly. The best known upper bound, proved by Spencer, Szemerédi and Trotter in 1984, is O(n^(4/3)), far above the linear guess but not contradictory.
OpenAI’s model, whose name and training details the company did not disclose, produced a construction proving that for some fixed improvement over linear, there are infinitely many n with ν(n) ≥ n^(1+δ). The model’s output was automatically graded before humans reviewed it. The final published proof, released as a PDF by OpenAI, is a human-edited exposition, not raw AI output. Separately, mathematician Will Sawin submitted an arXiv paper that refined the bound to an explicit ν(n) > n^1.014 for arbitrarily large n.
The companion arXiv paper, ‘Remarks on the disproof of the unit distance conjecture,’ was posted by Noga Alon, Thomas F. Bloom, W. T. Gowers, Daniel Litt, Will Sawin, Arul Shankar, Jacob Tsimerman, Victor Wang and Melanie Matchett Wood. They wrote that they had examined the AI-generated argument, verified its correctness, and then simplified and generalized it. The involvement of Bloom, who in 2025 had publicly challenged OpenAI’s earlier claims about GPT-5 solving Erdős problems, was particularly significant. His co-authorship signaled that this time the external check was rigorous and credible.
In October 2025, OpenAI promoted GPT-5 as having solved previously unsolved Erdős problems. Bloom and others quickly showed that the model had surfaced results already in the existing literature. OpenAI later acknowledged the misstep. That episode forced the company to pair its new claim with immediate, independent expert review.
The new result does not settle the planar unit-distance problem. The exact asymptotic growth of ν(n) remains unknown. The lower bound of n^1.014 is far below the 1984 upper bound of O(n^(4/3)), leaving a large gap. The proof has not been peer-reviewed and relies on an internal model whose architecture, training data and sampling process are not public. OpenAI described the solution as ‘the first autonomous AI solution of a major open problem in mathematics,’ a characterization that outside mathematicians have not endorsed independently, given that humans rewrote and strengthened the argument. Some discrete geometers outside the verification group have yet to comment publicly.
The result lands at a moment when AI is increasingly used in mathematical research. Other recent arXiv papers have applied large models to problem-solving and proof search, but this is perhaps the strongest example to date of a general-purpose model contributing a novel, independently verified solution to a classical problem. Even so, the workflow—AI output, automated grading, human verification, simplification and refinement—underscores that the tool still requires expert oversight at every stage.
For heads of research and strategy, the episode provides a calibrated signal: frontier AI can now generate original mathematical ideas that survive expert scrutiny, but the leap from a machine-generated counterexample to a fully autonomous mathematical reasoner has not been made. The value lies in the human-AI collaboration, not in replacing the human.