AI Helps Crack an 87-Year-Old Math Conjecture With One Tiny Formula

An AI-assisted counterexample disproves the Jacobian conjecture above two dimensions while leaving its original two-dimensional form open.

A formula short enough to fit inside a single post on X has sent a surge of excitement through the mathematical community, just as millions of people were settling down after the FIFA World Cup Final.

Levent Alpöge, a mathematician at the artificial intelligence (AI) company Anthropic, casually announced that he had found a counterexample to the Jacobian conjecture, a famous and long-standing problem in algebraic geometry. He reached the result using Anthropic’s large language model Claude Fable 5, which had been released to the public only weeks earlier.

Mathematicians working with large language models have recently produced several notable results. Yet this discovery stands apart from many of those earlier advances.

A simple question hides deep complexity

A conjecture is a mathematical statement that is thought to be true but has never been conclusively proved or disproved.

The Jacobian conjecture is highly abstract, but its central idea can be explained without advanced mathematics.

It concerns functions, which can be understood as small machines that receive one or more numbers and produce new numbers by following a particular equation or rule. The functions involved here are built from polynomials.

In this setting, the numbers can represent points in a space, similar to coordinates on a map. A function takes those coordinates and transforms them, effectively moving the points to different positions.

Mathematicians can examine how smoothly a function rearranges space by calculating its Jacobian determinant. When that determinant is always the same non-zero number, the function does not locally flatten, crush, or fold the space around any 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.

Decades of proofs failed under scrutiny

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.

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”.

One short formula changes the problem

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.

AI finds value beyond proving theorems

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.

Adapted from an article originally published in The Conversation.

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