An 87-year-old mathematical conjecture that generations of academics struggled to prove true has been disproven in a single social media post—thanks to a concise counterexample discovered with the assistance of artificial intelligence.
On July 19, Harvard University mathematician Levent Alpöge announced on X (formerly Twitter) that he had disproven the Jacobian conjecture, a legendary problem formally posed by German mathematician Ott-Heinrich Keller in 1939. The conjecture, which suggested that a specific type of mathematical function would automatically operate in reverse, was deemed so fundamental that Fields Medalist Stephen Smale listed it among 18 major mathematical challenges for the 21st century.
Rather than constructing a lengthy theoretical framework to validate the long-held assumption, Alpöge shattered it with a tiny counterexample spanning just 216 characters. The single line of code disproves the conjecture for three variables, though mathematicians note that a two-variable version of the riddle remains theoretically open.
In his post, Alpöge attributed part of the breakthrough to "fable"—an apparent reference to AI company Anthropic's Claude Fable 5 model—thanking the system for assisting him during the World Cup final. Mathematicians around the world were quick to verify Alpöge's compact counterexample, confirming its correctness despite widespread astonishment.
"People have been trying to prove it because it sounds, intuitively, very true," said Abhishek Saha, a mathematician at Queen Mary University of London. "I don't think that many people have been trying to disprove it. And now we have this one-sentence counterexample. This is probably the biggest conjecture that AI has played a significant role in so far."
The discovery marks a milestone in artificial intelligence's evolving relationship with higher mathematics. While previous AI successes involved checking proofs or resolving narrower sub-problems, delivering a clean counterexample to an 87-year-old cornerstone problem demonstrates unprecedented practical utility in advanced research.
However, scholars caution that the role of human insight remains far from obsolete. Chris Bowman-Scargill of the University of York highlighted that while AI can efficiently spot counterexamples, building major new theories and mathematical branches still demands human creativity. At the same time, Ivan Fesenko of Westlake University predicted that as AI models continue to rapidly advance, the landscape of mathematical research is on the brink of fundamental change.