Harvard mathematician Levent Alpöge has used AI to provide a counterexample to the Jacobian conjecture, a long-standing mathematical problem first proposed in 1939.

Levent Alpöge, a mathematician at Harvard University, announced on July 19 that he has disproven the Jacobian conjecture. Alpöge shared a 216-character counterexample on X, crediting his work to the AI model Claude Fable 5, developed by Anthropic. The conjecture, which suggested that specific mathematical functions could operate in reverse, had remained unproven since it was formally introduced by Ott-Heinrich Keller in 1939.

Significance of the discovery

Experts in the field consider this the most difficult mathematical problem yet solved with the assistance of AI. Abhishek Saha of Queen Mary University of London noted that while the result is surprising, the single-line counterexample is relatively easy to verify. The finding is significant because many mathematicians had intuitively believed the conjecture to be true, leading most research efforts toward proving it rather than disproving it.

AI's role in mathematical research

While the counterexample addresses the conjecture for three variables, researchers note that a two-variable version of the problem may still hold true. The use of AI in this discovery has prompted discussions regarding the future of the field. Chris Bowman-Scargill of the University of York highlighted that while AI can identify counterexamples, human creativity remains essential for building the complex, multi-page theories often required for major mathematical proofs. Ivan Fesenko of Westlake University suggested that increasingly capable AI models could lead to fundamental changes in how mathematics is practiced, potentially reaching the level of PhD-quality work within a year.