As millions of people were coming down from the excitement of the FIFA World Cup Final at the start of this week, a different kind of excitement was building within the mathematical community.

Levent Alpöge, a mathematician working at the artificial intelligence (AI) company Anthropic, made a very casual announcement on X that he had found a counterexample to the Jacobian conjecture, a very old and well-known problem in a field of mathematics called algebraic geometry. He had done this using Anthropic’s large language model Fable 5, released to the general public only a few weeks ago.

This is just the latest of many striking mathematical breakthroughs made by mathematicians working with large language models. But this one feels a little different to those that have come before.

What is the Jacobian conjecture?

First, what is a conjecture? It’s an idea that some mathematicians believe is true but nobody has been able to prove or disprove.

Now to the Jacobian conjecture. It’s fairly abstract but not too difficult to describe.

The conjecture involves functions, which are like little machines which you put one or more numbers into and out pop other numbers according to some rule or equation. In this case, the functions use what are called polynomials.

Specifically, it’s about situations where the numbers represent points in a space, like coordinates on a map. So we can imagine that when the function takes in some numbers and........

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