On May 20, 2026, OpenAI announced that one of its internal reasoning models had disproved a long-standing conjecture in discrete geometry connected to the planar unit-distance problem, first posed by Hungarian mathematician Paul Erdős in 1946. <cite index="7-1">The company said the result marks a milestone in AI-driven mathematics.</cite>
The problem
<cite index="7-2,7-3">The unit-distance problem is one of the best-known questions in combinatorial geometry, easy to state and remarkably difficult to resolve. The 2005 book Research Problems in Discrete Geometry, by Brass, Moser, and Pach, calls it "possibly the best known (and simplest to explain) problem in combinatorial geometry," and Princeton combinatorialist Noga Alon describes it as one of Erdős's favorite problems — Erdős even offered a monetary prize for its resolution.</cite> The problem asks: given n points placed in a plane, what is the maximum number of pairs that lie exactly one unit apart?
<cite index="7-5">Since Erdős's original work, the prevailing belief has been that "square grid" constructions were essentially optimal for maximizing the number of unit-distance pairs.</cite>
The result
According to OpenAI, its model produced an infinite family of constructions that beat the square-grid bound by a polynomial factor. <cite index="5-5">The construction draws on deep algebraic number theory — including Golod-Shafarevich theory and infinite class field towers — achieving n^(1+δ) unit-distance pairs for some fixed δ > 0, later refined to δ = 0.014 by Princeton mathematician Will Sawin.</cite>
The proof was reviewed prior to publication by external mathematicians. <cite index="4-4">University of Toronto mathematician Arul Shankar said in a statement provided by OpenAI that the model demonstrated "original, ingenious ideas" and was capable of carrying them out to fruition.</cite> <cite index="4-9,4-10">His colleague Jacob Tsimerman called it "an intimidating construction to see through," and noted that he had previously attempted to disprove the same conjecture without success.</cite> <cite index="3-3">Mathematician Misha Rudnev remarked that this was a problem he didn't expect to see solved in his lifetime.</cite>
Context and credibility
The announcement follows an earlier episode that drew scrutiny of OpenAI's mathematics claims. <cite index="6-4,6-5,6-6">Seven months earlier, the company's former VP Kevin Weil had posted on X that "GPT-5 found solutions to 10 (!) previously unsolved Erdős problems and made progress on 11 others" — but GPT-5 had in fact only located solutions that already existed in the literature, prompting taunts from rivals including Yann LeCun and Google DeepMind CEO Demis Hassabis, after which Weil took down the post.</cite>
The new result is being framed differently. <cite index="5-6">OpenAI describes it as the first time AI has autonomously solved a prominent open problem central to a subfield of mathematics.</cite> <cite index="1-10">The company said the work "offers an early glimpse of a new kind of collaboration between AI and human mathematicians."</cite>
Implications
The announcement intensifies an ongoing competition among frontier labs to demonstrate autonomous reasoning capabilities on tasks previously considered the exclusive domain of expert humans. It also sharpens debate around AI safety and oversight: a system that can generate original, verifiable mathematical arguments using sophisticated tools from algebraic number theory represents a qualitative shift from pattern-matching assistants toward systems capable of independent research contributions. External verification — a feature unique to mathematics, where proofs can be checked line by line — provided a guardrail in this case, but the broader question of how to evaluate autonomous AI claims in less verifiable domains remains open.
OpenAI has not disclosed which model produced the proof, describing it only as an internal, unreleased reasoning system.