OpenAI's Reasoning Model Disproves an 80-Year-Old Erdős Conjecture Using Number Theory

OpenAI on May 20, 2026 said one of its internal general-purpose reasoning models has disproved a central conjecture in discrete geometry that has stood since 1946 — the planar unit distance problem first posed by Paul Erdős. For nearly 80 years mathematicians had assumed that square-grid arrangements were optimal for maximising the number of pairs of points exactly one unit apart. The model found an infinite family of constructions that beats the grid, giving a polynomial improvement on the long-standing upper bound.

What sets this result apart from prior AI math wins is the route the model took. Instead of brute-forcing combinatorics, it reached for algebraic number theory, extending the classical Gaussian-integer approach into algebraic number fields and invoking infinite class field towers — heavy machinery that working mathematicians rarely connect to discrete geometry. The improvement in the exponent is small (roughly 0.014) but enough to settle the conjecture in the negative and shift the boundary of what is provable.

The proof has been reviewed by external experts, including Princeton number theorist Will Sawin and combinatorialists Noga Alon, Melanie Wood, and Thomas Bloom, who provided supportive statements. OpenAI researcher Noam Brown framed the speed of progress bluntly: "Less than 1 year ago frontier AI models were at IMO gold-level performance." Going from competition-style problems to disproving open conjectures in under a year is the part that has the research community paying attention.

Crucially, this was not a math-specialist system. The work came out of the same general-purpose reasoning stack OpenAI ships to ChatGPT and the API, not a model fine-tuned only on theorem proving. That is the signal markets and labs are reading: if a generalist agent can produce a publishable result in algebraic geometry by itself, the gap between AI-as-assistant and AI-as-collaborator in frontier science just narrowed sharply. Expect Anthropic, Google DeepMind, and xAI to point similar systems at their own list of open problems before the summer is out.