A Harvard mathematician has used artificial intelligence to disprove a long-standing problem in algebra that experts had spent decades trying to prove true. Levent Alpöge announced the finding on social media on 19 July with a short counterexample.
Alpöge posted a 216-character mathematical counterexample on X, showing that the Jacobian conjecture is false. The conjecture, proposed by Ott-Heinrich Keller in 1939, had also appeared on a list of major unsolved problems compiled by Stephen Smale in 1998.
The work involved Claude Fable 5, an AI model from Anthropic. Alpöge credited the system in his post for assistance during the World Cup final.
Abhishek Saha of Queen Mary University of London called the result the largest conjecture yet addressed with significant AI input. He noted that the counterexample works for three variables, though a two-variable version could still hold.
Chris Bowman-Scargill of the University of York and Ivan Fesenko of Westlake University said the development highlights AI's growing role in mathematics while underscoring that human insight remains essential for building new theories.