OpenAI says its AI proved a 1946 geometry conjecture
OpenAI says a general-purpose reasoning model cracked a 1946 Erdős geometry problem, and external mathematicians checked the proof.
OpenAI said a general-purpose reasoning model had disproved a geometry conjecture that has stood since 1946, a claim that turns a long-running test of mathematical ingenuity into a new stress test for AI credibility. The problem, the planar unit distance problem, asks how many pairs of n points in the plane can be exactly one unit apart. OpenAI said the proof was reviewed by external mathematicians and presented alongside a companion paper explaining the argument and why it matters.
The company said the model produced an infinite family of examples that improve on the square-grid constructions long treated as the benchmark for this kind of problem. In OpenAI’s telling, the gain is polynomial, not cosmetic, which is why the result has drawn attention well beyond artificial intelligence circles. The proof also uses ideas from algebraic number theory to solve an apparently elementary geometric question, a methodological twist that makes the result more striking than a brute-force search or a narrow symbolic trick.
Paul Erdős first posed the unit-distance problem in 1946, the same paper that introduced the distinct-distances problem. OpenAI’s proof paper says Erdős conjectured an upper bound of n^(1+o(1)) for the number of unit distances, while the best elementary upper bound remains O(n^(3/2)). Erdős reportedly valued the problem enough to offer a monetary prize for resolving it, which helped make it one of his favorite open questions in combinatorial geometry.
What makes OpenAI’s announcement unusual is not only the mathematics, but the verification. The company said the result came from a broad, general-purpose reasoning model rather than a system built specifically for mathematics or carefully scaffolded for this task. OpenAI called it the first time a prominent open problem central to a subfield of mathematics has been solved autonomously by AI. In a post defending the work, Fields Medalist Tim Gowers called it “a milestone in AI mathematics,” while number theorist Arul Shankar said current AI models can make “original ingenious ideas” and carry them “to fruition.”
The episode matters because it cuts against the idea that AI in advanced research is still limited to polished pattern matching. If the proof survives sustained scrutiny, it would suggest a narrower but more consequential role for frontier models: not replacing mathematicians, but helping generate and complete new arguments in places where human experts have spent decades pushing at the same boundary.
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