AI just disproved the 87-year-old Jacobian conjecture
On Sunday evening, as Spain and Argentina played out the World Cup final, the mathematician Levent Alpöge posted a short message on X. The Jacobian conjecture, he wrote, is false. He thanked a friend for asking about it, and another friend, “fable,” for working during the match.
That second friend was Fable 5, Anthropic’s latest AI model. Together they had produced a counterexample to a problem that had resisted mathematicians for 87 years. It was 216 characters long.
The Jacobian conjecture was set out by Ott-Heinrich Keller in 1939. In rough terms, it says that a certain kind of polynomial map, one whose Jacobian determinant is a non-zero constant, must be reversible with a neat polynomial inverse. It became one of the field’s most stubborn open problems. Stephen Smale put it on his famous 1998 list of challenges for the 21st century. New Scientist calls it the hardest maths problem an AI has yet cracked.
