‘Hello There the Jacobian Conjecture Is False Thanx’: Why a Tiny Social Media Post Has Mathematicians Rethinking AI

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

Levent Alpöge, a mathematician working at the artificial intelligence 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 Claude 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 puts out some other numbers, it is moving the points in space.

You can test how “nicely” a function moves everything around in space by calculating something called the Jacobian determinant. If the Jacobian determinant is always a constant number that is not zero, then the function never folds or crushes space around a particular point.

The Jacobian conjecture states that when the Jacobian determinant is a non-zero constant, there should always exist another function, also made up of polynomials, that reverses the original one. This will return all the points to their starting positions.

Not every function is reversible. For example, if our starting function moves two of the original points onto a single point, then we cannot reverse it. Once the points have been merged, we cannot distinguish between them to send them back to the right positions.

A Long History of Attempts—and Failures

The two-dimensional version of the Jacobian conjecture was stated by Czech mathematician Ludwig Kraus in 1884. It was generalized to any number of dimensions by German mathematician Ott-Heinrich Keller in 1939.

It was considered so compelling that Fields Medalist Stephen Smale included it in his 1998 list of Mathematical Problems for the Next Century.

During its long history, the Jacobian conjecture has been the subject of many claimed proofs, including by Beniamino Segre and Wolfgang Gröbner, two famed 20th-century mathematicians. However, in each case, subtle errors were found that invalidated the arguments.

Despite this, there have also been a number of valid efforts showing the conjecture is true with various restrictions. Computational results have also shown it is true in two dimensions for polynomials up to degree 100 (that is, including powers of the variables up to 100).

But nobody had proved the general case—or found an example showing the conjecture was wrong.