AI Counterexample Shatters 87-Year-Old Jacobian Conjecture

On 19 July, Harvard mathematician Levent Alpöge posted on X that the Claude Fable 5 system from Anthropic produced a 216-character counterexample to the Jacobian conjecture, a problem first formulated by Ott-Heinrich Keller in 1939. The brief construction was quickly verified by peers, showing that the conjecture—asserting that a certain class of polynomial maps is invertible—fails in three variables. Alpöge’s tweet credited “close friend fable” for the work; Anthropic declined comment.

New Scientist

The Jacobian conjecture has long been a benchmark of difficulty, appearing on Stephen Smale’s 1998 list of 18 “fiendishly difficult” problems for 21st-century mathematicians. Decades of effort focused on proving the statement true; few attempts aimed at disproving it. The new single-sentence counterexample overturns that trajectory.

New Scientist

Mathematics scholars note the result’s simplicity and the mystery of the prompting technique. Abhishek Saha (Queen Mary University of London) described the AI’s role as “the biggest conjecture that AI has played a significant role … so far in mathematics.” Chris Bowman-Scargill (University of York) cautioned that while AI can locate counterexamples, constructing extensive new theory—such as that required for Fermat’s Last Theorem—remains a human endeavor. Ivan Fesenko (Westlake University, China) projected that increasingly capable models may soon generate PhD-level work, prompting questions about the future need for mathematicians.

New Scientist

The episode highlights a shift: AI can now produce concise, verifiable mathematical statements, potentially accelerating the resolution of open problems. Yet scholars emphasize that deeper theoretical development and the creation of new mathematical frameworks still rely on human creativity. The community anticipates further AI contributions while debating long-term impacts on research practice and professional roles.

New Scientist

• “Probably this is the biggest conjecture that AI has played a significant role [in proving or disproving] so far in mathematics,” — Abhishek Saha, Mathematician

New Scientist

• “I don’t know how he did it, what exactly was the prompt to give Fable, because if one were to search everything, it wouldn’t quite work, so obviously there was some insight also which is not currently published.” — Abhishek Saha

New Scientist

• “If you look at Fermat’s last theorem [which was solved by Andrew Wiles in 1994], you had to create a hundred pages of new mathematics – you had to build a whole big theory in order to solve a conjecture,” — Chris Bowman-Scargill, Mathematician

New Scientist

• “Right now, AI can already produce master’s degrees in mathematics. In one year, they will produce PhD degrees in mathematics. And then the question arises, do we really need so many mathematicians around if AI can do such things so nicely?” — Ivan Fesenko, Mathematician

New Scientist