As millions of people were coming down from the excitement of the FIFA World Cup Final recently, 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 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 generalised to any number of dimensions by German mathematician Ott-Heinrich Keller in 1939.
It was considered so compelling that Fields Medallist 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.
A deceptively simple answer
One of the key reasons the Jacobian conjecture is so intriguing is that, in theory, it should be easy to find a counterexample. It is straightforward to come up with examples of functions that merge points, and also examples of polynomial mappings that have a constant Jacobian determinant.
However, finding a polynomial mapping with both properties is the challenge. Indeed, as one Math Stack Exchange user noted in a post from 2017, “for all what we know, some smart undergraduate can simply write a formula […] that will be a counter-example to this conjecture”.
Indeed, this did turn out to be the case for Alpöge’s function, which is short enough to fit into a single X post. He found an example of a function in three dimensions which has a constant Jacobian determinant of -2, and which moves multiple input points to the same output point, so it is not reversible.
It shows the conjecture is false for every dimension larger than 2, with the original conjecture in two dimensions remaining open. The brevity of the counterexample made it easy for other mathematicians to verify.
The latest advance in a growing series
Alpöge’s discovery is the latest in a string of high-profile mathematical breakthroughs made by large language models. Recent examples include OpenAI’s disproof of the unit distance conjecture, and the proof of Erdős’ problem 1196 by Liam Price, a 23-year-old amateur mathematician.
Both examples illustrate one of the most striking strengths of AI models. They can draw on ideas from different areas of mathematics, combining them in a novel way to prove astonishing results.
At the time of writing, details have not been made public regarding exactly how Alpöge prompted the AI model to produce the Jacobian conjecture counterexample and what its output looked like. However, so far this result appears to be of a different nature.
Unlike many other recent AI-assisted breakthroughs, the counterexample itself is remarkably simple. The difficulty in finding it seems to have lain not in an intricate construction or a lengthy proof, but rather in finding a good way of navigating an enormous search space of possible polynomial mappings to find one with the right properties.
This suggests AI may prove to be just as valuable for discovering unexpected mathematical objects as it is for constructing proofs. What this means for the future of mathematics – and human mathematicians – remains to be seen.
