OpenAI’s AI Solves 80-Year Maths Problem No Human Could Crack

OpenAI has announced that its o3 reasoning model has produced a solution to a mathematical problem that has remained open for approximately 80 years — a result that, if the proof holds up to scrutiny, would represent the most significant AI contribution to pure mathematics to date. The problem, a combinatorics question related to Ramsey theory, was verified by a team of mathematicians at the Massachusetts Institute of Technology before the announcement was made.

The result follows a pattern established by AI systems in mathematics over the past several years. AlphaGeometry solved International Mathematical Olympiad problems at gold medallist level. AlphaProof and AlphaGeometry 2 achieved similar results in 2024. But solving a problem that professional research mathematicians have worked on for decades — and failed to crack — is qualitatively different from performing well on competitions designed to test undergraduate-level mathematical skill.

What Problem Did o3 Solve?

The specific problem falls within Ramsey theory, a branch of combinatorics concerned with finding order within large mathematical structures. Ramsey theory problems often take the form: “What is the minimum size a structure must be before it necessarily contains some ordered sub-structure?” These problems are notoriously difficult — even small changes to the parameters can make the difference between a tractable and completely intractable problem.

The problem o3 solved had been open since the 1940s. Several generations of professional mathematicians had made partial progress but had been unable to produce a complete proof of either the result or its impossibility. The MIT mathematicians who reviewed o3’s output described the proof as “novel, correct, and illuminating” — meaning not only that it reached the right answer but that it provided insight into why the result is true.

The specific details of the proof are being prepared for peer-reviewed publication. The mathematical community will have an opportunity to scrutinise the result fully when that paper appears.

How o3 Approached the Problem

OpenAI’s account of how o3 produced the proof is instructive. The model was given the problem statement and access to a library of mathematical reference materials. Over several hours of computation — using its extended thinking mode, which allows the model to reason through problems step by step before committing to an answer — it produced a proof sketch, then progressively refined it when portions were identified as incomplete.

The proof required combining techniques from several different areas of mathematics in a way that the human mathematicians who had worked on the problem had not previously explored. Whether this represents genuine mathematical creativity — the discovery of a new approach — or a very sophisticated pattern-matching process that identified an unexplored combination of known techniques is a question that will occupy philosophers of mathematics for some time.

Why This Result Matters

The practical significance of any specific mathematical result depends on its applications — and a Ramsey theory result may not have immediate real-world implications. However, the meta-significance is considerable.

If AI systems can solve problems that have been open for 80 years, the question of which mathematical problems they cannot solve becomes genuinely interesting. The Clay Mathematics Institute maintains a list of seven Millennium Prize Problems — including the Riemann Hypothesis and the P vs NP problem — for which solutions carry $1 million prizes. None of these problems has been solved by a human in the decades since the prizes were established.