An 87-Year-Old Puzzle
Imagine a complex mathematical machine that transforms numbers. The Jacobian conjecture, first proposed by German mathematician Ott-Heinrich Keller in 1939, asked a seemingly simple question: if this machine works in a very specific, neat way, can you
always create another machine that perfectly reverses its work? For nearly nine decades, no one could provide a definitive yes or no. The problem was so fundamental and famously difficult that it was included on a prestigious list of essential unsolved problems for the 21st century. Mathematicians around the globe tried and failed to prove it, and it became notorious for attracting incorrect proofs. The prevailing belief was that the conjecture was likely true, but the proof remained just out of reach, a tantalizing mystery at the heart of algebraic geometry.
Enter the AI Collaborator
The breakthrough came not from a dusty chalkboard but from a dialogue between a human and a machine. Levent Alpöge, a mathematician at Harvard University with ties to the AI company Anthropic, used their latest model, Fable 5, to attack the problem. Instead of trying to prove the conjecture true for all cases, the AI was tasked with a different mission: find just one instance where it fails. This is known as finding a counterexample. In mathematics, a single valid counterexample is enough to disprove a long-standing conjecture entirely. The AI succeeded, producing a 216-character string of polynomial equations that meets all the criteria of the conjecture, yet cannot be reversed. It was the needle in a haystack that humans had been searching for since the start of the Second World War.
A Tweet Shakes the Math World
The reveal was fittingly modern. Alpöge announced the discovery in a short post on social media platform X, reportedly while the world was distracted by the World Cup final. The reaction from the mathematical community was immediate and electric. Leading mathematicians expressed their astonishment, with one Fields medallist noting it was the first time an AI had solved a major problem he had even heard of. The counterexample was quickly verified by other experts, both by hand and using formal proof-checking software, confirming that the 87-year-old conjecture was, in fact, false. The moment highlighted a dramatic shift, where a problem that had occupied generations of human thought was dismantled by an AI in a matter of moments.
A New Partner in Discovery?
This achievement has sparked a fascinating debate about the role of AI in scientific research. Is the machine truly 'thinking', or is it just performing an incredibly sophisticated search? Some experts argue that finding a counterexample is precisely the kind of brute-force search problem that computers are destined to be good at. They contrast this with other recent AI achievements in mathematics, like the disproof of the Erdős unit distance problem by an OpenAI model in May 2026, which was seen as a more creative breakthrough that revealed new, unexpected connections between different fields of math. The Jacobian counterexample, while monumental, is viewed by some as less of a grand theoretical insight and more of an answer that ends an argument. It doesn't necessarily teach mathematicians why the conjecture is false in a broader sense, but it definitively proves that it is. This distinction is crucial, pointing toward a future where AI acts as a powerful collaborator, capable of hunting for answers in vast spaces of possibility that are too large for the human mind to explore on its own.
