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An internal OpenAI reasoning model has disproved the Erdős unit distance conjecture, a problem in combinatorial geometry that sat open since Paul Erdős posed it in 1946. The result is a first: AI autonomously solving a prominent open problem central to a field of mathematics, not a competition problem with known solutions somewhere in the literature.

Timothy Gowers, Fields Medal winner, said he would recommend the proof for publication in the Annals of Mathematics “without any hesitation.” Three external mathematicians reviewed and signed off: Noga Alon, Melanie Wood, and Thomas Bloom, who maintains the Erdős Problems website.

The Problem

The planar unit distance problem is simple to state. Place $n$ points anywhere in the plane. Count how many pairs sit at distance exactly 1 apart. What placement maximizes that count?

For nearly 80 years, the best known lower bounds came from square lattice constructions. Erdős conjectured that the answer grows as $n^{1+O(1/\log\log n)}$ — meaning grids were essentially optimal. The best known upper bound, $O(n^{4/3})$, left a gap that no approach had closed.

The OpenAI model broke that picture. It found an entirely new family of constructions, drawn from a higher-dimensional Minkowski lattice projected to the plane using CM number fields. The construction produces point sets with more than $n^{1+\delta}$ unit distance pairs for some $\delta > 0$ — a polynomial improvement over the grid baseline, disproving Erdős’ conjecture.

How It Was Done

The proof was produced by Lijie Chen using an internal OpenAI model and verified for correctness by mathematicians Mark Sellke and Mehtaab Sawhney. The summarized chain of thought runs 125 pages.

The key ingredient is a bridge from discrete geometry into algebraic number theory: specifically, class field theory and Golod-Shafarevich towers. The model constructed sequences of CM fields with large degree and small discriminant, where many primes split in a controlled way. This produces lattices with many pairs of vectors projecting to unit distance in the plane. The number-theoretic machinery is far from the surface of the problem — it had not appeared in unit distance research before.

Noam Brown, OpenAI research scientist, confirmed: this is a general-purpose LLM. Not a theorem prover, not a model targeted at mathematics. The same class of model used for code review and document summarization connected algebraic number theory to combinatorial geometry because it had enough test-time compute to hold the chain of ideas together.

What It Means

A mathematician-level endorsement for Annals of Mathematics from a Fields Medalist is not a routine benchmark result. The gap between “useful coding assistant” and “original research contributor in pure mathematics” has closed faster than the field expected.

Fifteen Erdős problems have moved from open to solved since January 2026. Eleven of those are credited to AI. In October 2025, OpenAI made a false claim about 10 solved problems that turned out to be rediscovered existing results. This time the company published the full proof document, named the external validators, and waited for written endorsement before announcing.

Will Sawin, Columbia University mathematician, subsequently published an explicit version of the construction that pins the exponent at $n^{1.014}$, compared to the inexplicit $n^{1+\delta}$ of the original. The original conjecture narrowing the gap to the upper bound $O(n^{4/3})$ now sits as the next open problem. Sawin’s paper notes the method is almost certainly not optimal.

Terence Tao noted earlier this year that ideas were becoming cheap while verification remained expensive. This result shifts the balance: the model produced an idea that human mathematicians had not found in eight decades, and external verification confirmed it was correct.