OpenAI Challenges Erdős: An AI Solves a Historic Math Problem

A Unique Mathematical Breakthrough Thanks to OpenAI's AI

A reasoning model developed by OpenAI has recently succeeded in disproving a conjecture posed by the famous Hungarian mathematician Paul Erdős in 1946. This advancement has been validated by nine independent mathematicians, and Fields Medalist Tim Gowers has deemed the result worthy of publication in a major scientific journal.

The model approached the problem by stepping away from the initial domain of geometry to explore algebraic number theory, an unexpected yet fruitful approach. Erdős sought to determine how many pairs of points could remain at the same distance on a plane, proposing a conjectural upper bound. OpenAI surpassed this limit by discovering an infinite family of arrangements, with a gain quantified by Will Sawin from Princeton at an exponent of approximately 0.014.

A Generalist Model Serving Mathematics

What makes this achievement even more remarkable is that the model used by OpenAI was not specifically trained for mathematics. It is the same type of model that can draft emails or produce Python code. Sebastian Bubeck and Noam Brown, researchers at OpenAI, emphasized that this model has nothing to do with AlphaProof from Google DeepMind, which is designed for formal proof.

The model was able to discard classical avenues to focus on an innovative approach, leveraging high-degree algebraic numbers to construct a counterexample. The complete reasoning, which spans 125 pages, was initially met with skepticism by Sebastian Bubeck, given how promising it seemed.

Supervision and Validation by Experts

Despite the model's autonomy, three researchers from OpenAI ensured supervision and verification of the process. Lijie Chen led the model, while Mark Sellke and Mehtaab Sawhney verified the correctness of the reasoning. Both Sellke and Sawhney are recognized mathematicians, having received prestigious awards in the field.

However, some reviewers expressed reservations, as no external examiner consulted the raw output of the model. The published version was edited and refined with the help of Codex for clarity. OpenAI maintains that the proof file was generated without human intervention on the mathematical content.

A Precedent and Future Perspectives

OpenAI had previously encountered difficulties in October 2025, when Kevin Weil, the former vice president of the company, mistakenly announced that GPT-5 had solved ten Erdős problems. This time, Thomas Bloom, who had criticized the previous announcement, co-signs the verification paper, highlighting the significance of this achievement.

Although the conjecture has been disproven, the exact value of the maximum remains unknown, and the best known upper bound still belongs to Szemerédi. The peer review process will take several more months, but Tim Gowers has expressed confidence in the quality of the work for publication in the Annals of Mathematics.

The Details of the Proof and the Involvement of Mathematicians

In its reasoning, the model dismissed several classical avenues, ranging from roots of unity to powers of a rational point on the circle. The decisive shift came from an unexpected observation: any optimal arrangement can be described with algebraic numbers, although this involves a gigantic degree. The model recognized this high degree as an opportunity for a counterexample, where others would have seen only a complication.

Sebastian Bubeck shared that the model's first output seemed too good to be true. The summary of the reasoning spans 125 pages, reflecting the complexity and depth of the analysis performed by the AI.

Verification and Experts' Reservations

According to OpenAI, no AI had ever independently solved an open problem of such significance. However, the autonomy of language models has its limits. Three researchers played a key role as safeguards. Lijie Chen piloted the internal model, while Mark Sellke and Mehtaab Sawhney verified its correctness. All three are recognized figures in the field of mathematics, with distinctions such as the Morgan Prize and medals from international mathematics Olympiads.

Several reviewers raised concerns, noting that no external examiner had access to the raw output of the model. The disseminated version was edited and refined with Codex for clarity. Nevertheless, OpenAI asserts that the proof file was generated in one go, without human intervention on the mathematical content.

In the companion paper, the nine mathematicians linked the ideas to references such as Ellenberg-Venkatesh, Golod-Shafarevich, and Hajir-Maire-Ramakrishna, which the model does not directly cite. Will Sawin even simplified the argument after reading it, reducing the construction to a single rational prime.

A Precedent and Future Perspectives

OpenAI had already stumbled in this area in October 2025. At that time, former vice president Kevin Weil announced that GPT-5 had solved ten Erdős problems, while the model had merely rediscovered already published solutions. Thomas Bloom, who maintains the site cataloging Erdős problems, denounced this distortion of facts, and Weil retracted his message. This time, Bloom co-signs the verification paper, asserting that no AI has ever achieved such a performance in mathematics.

Erdős placed great importance on his favorite questions, offering $500 to anyone who could solve this upper bound, which is approximately 430 euros. The model has disproven the conjecture, but the exact value of the maximum remains unknown, and the best known upper bound still belongs to Szemerédi.

External teams outside the initial panel of reviewers will read the proof in the coming weeks, while the complete peer review process will take months. Tim Gowers wrote that a reporter from the Annals of Mathematics would accept the text without hesitation, but the proof has not yet been submitted to that journal.