A longstanding mathematical problem known as the Jacobian conjecture, which has challenged mathematicians for more than eight decades, may have been disproven by a junior fellow at Harvard University with the assistance of artificial intelligence. Levent Alpöge, 33, shared a concise mathematical expression on the social media platform X, accompanied by a statement asserting that the conjecture is false.

The Jacobian conjecture, formulated in 1939, concerns the invertibility of certain polynomial functions in multiple variables. It posits that if such a function’s Jacobian determinant—a measure related to how the function distorts space—is a nonzero constant, then the function should have a polynomial inverse, effectively allowing one to "reverse" the function to its original state without distortion. Despite numerous attempts over nearly 90 years, no universally accepted proof or disproof has been found.

Alpöge’s post consisted of a 216-character string representing a specific polynomial function in three complex variables, along with its Jacobian determinant and several coordinate mappings. He credited an AI model known as Claude Fable 5, developed by Anthropic, a company specializing in AI safety and research, for helping arrive at the solution. In his message, Alpöge called the conjecture false, providing an explicit counterexample through the function he described.

Experts have reacted with cautious interest. Abhishek Saha, a mathematician at Queen Mary University of London, praised the work as significant and noted that several mathematicians had verified Alpöge’s calculations. However, he emphasized that the precise manner in which the AI model contributed to the discovery remains unclear, including the specific prompts and interactions that guided Alpöge’s reasoning. Saha pointed to a similar recent achievement in which AI helped solve a conjecture posed by Paul Erdős, highlighting the increasing role of machine assistance in solving complex mathematical problems.

Alpöge holds advanced degrees in mathematics from the University of Cambridge and Princeton University. When approached for comment, he did not provide further details beyond the information shared in his initial post. As mathematicians continue to study and validate the findings, the potential overturning of the Jacobian conjecture stands as a notable milestone in the interplay between human insight and artificial intelligence in mathematical research.