Tim Gowers: What sort of maths are LLMs good at?

Gowers is writing a few days after OpenAI announced it had solved ten major problems in mathematics and theoretical computer science, including the first construction of a non-sofic group and a proof that the multicolour Ramsey number (with 3 colors) grows superexponentially. He notes these were among the most important unsolved problems in group theory and Ramsey theory respectively, and the post is intended as a snapshot of the situation in early August 2026.

Despite these impressive results, Gowers doesn't think LLMs are yet better than all humans at all aspects of mathematics; if they were, their speed advantage would produce a much larger flood of results. He's interested in identifying what kind of problems LLMs are good at and where they still need improvement, even though he admits he doesn't have a crisp classification.

Gowers considers the theory that LLMs are particularly good at finding counterexamples. He observes that LLMs can also find proofs, but most of their most famous solved problems—including the non-sofic group, the Ramsey number result, the Jacobian conjecture, and the unit distance conjecture—are counterexamples rather than proofs.

To make the counterexample theory convincing, Gowers says two things are needed: first, to decide when solving a problem counts as finding a counterexample, and second, to explain why LLMs would be particularly suited to that kind of problem. He then points out that the notion of "finding a counterexample" is not as obvious as it might seem.