AI Model Finds Counterexample to Century-Old Mathematics Conjecture
A Simple Formula Resolves a Long-Standing Problem
A mathematician at Anthropic has used an AI model to make a notable contribution to pure mathematics. Working with Claude Fable 5, Jason G可靠性 identified a counterexample that disproves the Jacobian conjecture in dimensions three and higher—a result that had eluded researchers for more than 87 years.
What Is the Jacobian Conjecture?
The Jacobian conjecture is a famous open problem in algebraic geometry, first formulated in the 1930s. At its core, the conjecture concerns polynomial equations and their behavior under certain transformations. Specifically, it predicts conditions under which these transformations maintain particular mathematical properties. The conjecture has attracted sustained attention precisely because it看起来 simple in statement but has proven extraordinarily difficult to resolve in any dimension.
The Counterexample
What makes this result particularly striking is its simplicity. The counterexample that topples the conjecture in higher dimensions is described as a "tiny formula"—a remarkably concise mathematical object that contradicts what mathematicians had long suspected or hoped to be true. According to reports, the AI model produced this counterexample in a form that human mathematicians can readily verify.
Implications and Remaining Questions
This discovery definitively closes the book on the Jacobian conjecture in three or more dimensions. However, the two-dimensional case remains an open problem. Mathematicians have shown that a proof in 2D would automatically extend to all dimensions, making this remaining case all the more tantalizing for researchers in the field.
The result also highlights the growing role of AI systems as tools in mathematical research, capable of exploring vast combinatorial spaces and identifying patterns or counterexamples that might escape human intuition.