AI helps mathematician disprove 87-year-old conjecture
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- Levent Alpöge at Harvard University tweeted on 19 July that the Jacobian conjecture — first posed by Ott-Heinrich Keller in 1939 and later included on Stephen Smale's 1998 list of 18 hard problems for the 21st century — is false, posting a 216-character counterexample that mathematicians have already verified.
- Alpöge credited his "close friend fable," Anthropic's Claude Fable 5, for part of the work, noting the AI was running during the World Cup final; Anthropic declined to comment.
- Abhishek Saha at Queen Mary University of London called the result "the biggest conjecture that AI has played a significant role in" proving or disproving so far in mathematics, noting the solution is simple to check but the path to it is not published.
- The counterexample disproves the three-variable version of the conjecture, but a two-variable variant could theoretically still hold, leaving open mathematical questions.
- Chris Bowman-Scargill at the University of York drew a distinction between AI finding counterexamples and the century-spanning work of human mathematicians like Andrew Wiles, who built "a hundred pages of new mathematics" to solve Fermat's Last Theorem.
- Ivan Fesenko at Westlake University in China argued AI already produces master's-level mathematics and will produce PhD-level work within a year, asking "do we really need so many mathematicians around" as the field faces "fundamental change."
Why it matters: An 87-year-old conjecture that mathematicians spent decades trying to prove was instead disproved with a one-line counterexample produced with AI help, surprising a field that hadn't seriously attempted a refutation. The result signals that AI-assisted mathematics has crossed a threshold in the prestige and difficulty of problems it can touch — and Fesenko's warning that AI could produce PhD-level math within a year frames a direct labor-market question for academic mathematicians.
