Mathematician posts 216-character counterexample disproving Jacobian conjecture for three variables

Levent Alpöge announced on X on 19 July that the Jacobian conjecture is false, posting a 216-character counterexample as proof. The conjecture, formally set out by Ott-Heinrich Keller in 1939, had remained unresolved for nearly nine decades and appeared on Stephen Smale’s 1998 list of 18 difficult problems for 21st-century mathematicians.

Alpöge credited part of the work to his “close friend fable,” referring to Anthropic’s Claude Fable 5 AI, and thanked the system for working during the World Cup final.

New Scientist reported that the single line of mathematics was simple to verify and has already been confirmed by many mathematicians. The counterexample disproves the conjecture for three variables, though a version with two variables could still hold.

Abhishek Saha at Queen Mary University of London stated that this is probably the biggest conjecture AI has played a significant role in proving or disproving so far, adding that AI has made remarkable progress in the last year.

Chris Bowman-Scargill at the University of York noted that finding counterexamples differs from building whole new branches of mathematics, which still requires human creativity. He contrasted the result with proofs such as Fermat’s last theorem that demanded extensive new theory.

Saha said people had largely tried to prove the conjecture because it sounded intuitively true, making the one-sentence disproof a surprise.

The finding leaves open questions about how the AI was prompted and whether further cases remain.