OpenAI announced on May 20 that an internal reasoning model had produced a proof refuting a central conjecture in the planar unit distance problem, an 80-year-old question posed by Paul Erdős. The claim marks one of the strongest demonstrations of an AI system autonomously generating an original mathematical argument, and it arrives with an unusual endorsement: a group of external mathematicians, including a past critic, verified the core result.
That stands in sharp relief to OpenAI’s October 2025 claim that GPT-5 had solved 10 Erdős problems, which unraveled after researchers showed the model had simply retrieved existing solutions. The episode signals that AI reasoning may occasionally yield surprising insights, but it also highlights the critical role of independent human verification.
The unit distance problem asks for the maximum number of point pairs exactly one unit apart among n points in the plane. The widely held conjectured bound, assumed for decades, was that the maximum grows only a little faster than linearly—precisely, n^(1+o(1)). The new proof shows that for infinitely many n, there are at least n^(1+δ) such pairs for some positive δ, contradicting that assumption. The proof does not determine the true asymptotic maximum; the best known upper bound remains O(n^(4/3)). A $500 prize, offered for a full solution, remains unclaimed.
OpenAI said the proof was generated by an unnamed general-purpose reasoning model, not a math-specialized system, via an AI-written prompt and an automated grading pipeline before human examination and editing. The company published the proof and a companion remarks paper on May 20, but the work has not yet undergone formal peer review.
The companion paper—authored by Noga Alon, Thomas Bloom, W. T. Gowers, Daniel Litt, Will Sawin, Arul Shankar, Jacob Tsimerman, Victor Wang and Melanie Matchett Wood—confirmed and strengthened the AI’s counterexample. Bloom had been a prominent critic of the earlier GPT-5 claims, and his involvement alongside other leading mathematicians gives this result unusual credibility for an AI-generated proof. Sawin separately posted an explicit lower bound exceeding n^1.014 unit-distance pairs, providing the first concrete exponent.
Yet the model’s architecture, training data, and failure rate remain undisclosed, and the exact extent of human editing is not public. The proof has not been reviewed by a journal, and it is unclear whether other models could replicate the result.
For AI research, the episode demonstrates that a general-purpose reasoning system can occasionally produce a novel mathematical argument that withstands expert scrutiny—a step beyond pattern-matching. Unlike systems that remix known proofs, the model generated a construction using algebraic number theory that experts considered original. Still, it remains a single demonstration from a single company, with no auditable pipeline. The proof now heads for broader community examination and eventual journal review. OpenAI’s decision to publish the proof and companion paper sets a higher standard after its previous overreach, but the gap between a single striking result and a reliable tool for mathematics remains wide.