Levent Alpöge, a mathematician at artificial intelligence company Anthropic, announced the discovery of a counterexample to the Jacobian conjecture using Claude Fable 5, the company’s large language model that became publicly available only weeks prior. The finding represents a significant development in mathematical research, as the conjecture had remained unresolved since its formulation in the late 19th and early 20th centuries.
The Jacobian conjecture concerns polynomial functions that map points in multidimensional space. It proposes that if a function preserves a specific mathematical property called the Jacobian determinant—meaning it never folds or crushes space—then another polynomial function should exist that reverses it. The conjecture originated with Ludwig Kraus in 1884 for two-dimensional cases and was generalized by Ott-Heinrich Keller in 1939 to any number of dimensions. The problem gained additional prominence when Fields Medallist Stephen Smale included it in a 1998 list of important unsolved mathematical problems.
Despite numerous attempted proofs over the decades, including efforts by renowned mathematicians, the general case remained unproven. Computational verification confirmed the conjecture’s validity for two-dimensional polynomials up to degree 100, and various restricted versions were proven true. However, the discovery of a definitive counterexample or a universal proof eluded researchers. Alpöge’s counterexample demonstrates that the conjecture is false for all dimensions greater than two, leaving the original two-dimensional case still open. The function he found is remarkably compact—brief enough to fit in a social media post—which enabled rapid verification by other mathematicians.
The breakthrough exemplifies how large language models are increasingly contributing to mathematical discovery. Recent years have seen AI-assisted proofs of previously unsolved problems, including OpenAI’s disproof of the unit distance conjecture and solutions to other long-standing mathematical questions. Alpöge’s discovery differs notably from many other AI-assisted breakthroughs in that the counterexample itself is elegantly simple rather than involving intricate constructions or lengthy proofs. The primary challenge appears to have been navigating an enormous search space of possible polynomial mappings to identify one with the required properties, a task the AI model performed effectively.
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