- Anthropic’s AI model Claude Fable 5 helped disprove the Jacobian Conjecture from 1939
- Mathematician Levent Alpoge used the AI during the FIFA Club World Cup final to find a counterexample
- The AI-generated polynomial formula had a constant Jacobian but was not reversible, disproving the conjecture
Artificial intelligence has achieved one of its biggest mathematical breakthroughs yet. Anthropic's AI model, Claude Fable 5, recently helped disprove the Jacobian Conjecture, an unsolved problem in algebraic geometry that had puzzled mathematicians since 1939. The breakthrough was made by mathematician Levent Alpoge, a former Harvard researcher who now works at Anthropic.
On Sunday night, while watching the FIFA Club World Cup final, Alpoge asked Claude Fable 5 to work on the Jacobian Conjecture after a friend, Akhil, brought up the problem. Instead of supporting the conjecture, the AI produced something mathematicians had been searching for unsuccessfully for nearly 90 years - a counterexample showing the conjecture was false. Alpöge later shared the result on X, posting a 216-character mathematical formula that demonstrated the finding. The tweet has close to 30 million views.
"Hello there, the Jacobian conjecture is false. Thanks to my close friend Akhil for asking about it and my other close friend Fable for working during the World Cup final," he wrote.
Check out the tweet here:
What is the Jacobian Conjecture?
Proposed by German mathematician Otto Keller in 1939, the Jacobian Conjecture claimed that a certain type of polynomial map with a constant, non-zero Jacobian determinant must always be reversible.
The problem became one of the best-known unsolved questions in algebraic geometry and was included in mathematician Stephen Smale's list of major open mathematical problems in 1998.
How did the AI disprove it?
Claude Fable 5 generated a simple polynomial formula involving three variables.
The formula had the required constant Jacobian determinant of -2, but it also showed that three different inputs produced exactly the same output. Because different inputs led to the same result, the mapping could not be reversed - directly contradicting the Jacobian Conjecture.
Leading mathematicians, including Terence Tao and Jared Duker Lichtman, reviewed the counterexample and confirmed that it successfully disproves the long-standing conjecture.
The two-dimensional version of the problem, however, remains unsolved.
AI's growing role in mathematics
This is not the first major mathematical breakthrough involving AI. In May, OpenAI announced that one of its internal research models had also found a counterexample to the Erdos Unit Distance Conjecture, another famous open problem.
Researchers say these advances show that AI can explore unconventional lines of reasoning that humans might overlook.
