AI Takes Down a 90‑Year‑Old Math Puzzle

When a piece of software suddenly claims to have solved a problem that has puzzled mathematicians for almost a hundred years, it feels a bit like watching a magician pull a rabbit out of a hat – you’re delighted, but you also want to check whether the rabbit is real.

That’s the situation Levent Alpöge, a mathematician who works in algebraic geometry, found himself in last week. In a surprisingly candid post on X (formerly Twitter), he announced that he had used Anthropic’s new language model, Fable 5, to produce what appears to be a genuine counterexample to the Jacobian conjecture.

The Jacobian conjecture, first posed in 1939, asks a deceptively simple question: if a polynomial map F from ℂⁿ to ℂⁿ has a Jacobian determinant that is identically 1, must F necessarily have a polynomial inverse? In plain English, the conjecture says that a certain kind of “smooth” polynomial function should always be reversible by another polynomial function. For decades the conjecture has been listed among the most stubborn open problems, even landing on Steve Smale’s famous list of 18 problems for the 21st century in 1998.

Alpöge’s tweet – posted on July 23, 2024 – included a link to a short note in which he described the steps Fable 5 took. The AI was fed the formal statement of the conjecture, a handful of known partial results, and asked to search for a map that satisfies the Jacobian‑determinant condition but fails to be invertible. After a few rounds of prompts, the model produced an explicit polynomial map in three variables, together with a detailed computation showing that its Jacobian determinant is indeed 1 everywhere, yet the map does not admit a polynomial inverse.

"I ran the calculations myself and they check out," Alpöge wrote, adding that he had double‑checked the algebra with a computer‑algebra system. "If this holds up, it means the Jacobian conjecture is false." He emphasized, however, that the result is still provisional. "Extra eyes are needed – other experts should verify the steps, because a single slip could resurrect the conjecture," he cautioned.

The math community’s reaction has been a mixture of excitement, skepticism, and a healthy dose of curiosity. Some senior algebraic geometers have praised the breakthrough as a potential turning point, noting that AI‑generated insights have already helped with conjecture formulation and pattern spotting. Others warned that history is littered with claimed counterexamples that later turned out to rely on hidden assumptions or computational errors.

What makes this episode especially intriguing is the way the AI was used. Rather than handing over a finished proof, Alpöge treated Fable 5 as a collaborative partner: he iteratively refined the prompts, asked for intermediate simplifications, and let the model explore a vast space of polynomial constructions that would be infeasible for a human to test manually. The result is a hybrid workflow that blends human intuition with the brute‑force searching power of modern large language models.

Whether the Jacobian conjecture is finally laid to rest will be decided in the weeks and months ahead, as other researchers attempt to reproduce and extend Alpöge’s calculations. If the counterexample survives scrutiny, it will be a landmark moment: the first time a deep, long‑standing problem in pure mathematics was solved – or rather, disproved – by an artificial intelligence.

For now, the mathematical world watches with bated breath, aware that we might be witnessing the start of a new era where machines not only assist in calculations but also help uncover fundamental truths that have eluded us for generations.