AI Cracks an 87-Year-Old Math Riddle

Mathematicians have spent decades trying to prove the Jacobian conjecture was true. It turns out it was false, and a 216-character counterexample was all it took to show it. The person who found it credited an AI assistant for part of the work.[1]

The Jacobian conjecture, formally set out by Ott-Heinrich Keller in 1939, suggests that a certain type of mathematical function would also work in reverse. It made Stephen Smale's influential list of 18 fiendishly difficult problems for the 21st century, drawn up in 1998. For 87 years, the conjecture held. Most people working on it were trying to prove it true. Almost nobody was trying to disprove it.[2]

On July 19, a mathematician posted the counterexample on social media. A single line of mathematics, simple enough to verify, that showed the conjecture was false with three variables. The announcement was unceremonious. He thanked his "close friend fable" for working during the World Cup final, seemingly referring to an AI model from a major AI lab.[1]

Easy to Check, Hard to Find

Experts say this is the most difficult mathematical problem yet solved with AI assistance. What makes it remarkable is the gap between difficulty of finding the answer and ease of checking it. The counterexample is a single line. Many mathematicians have already verified it. But nobody knows exactly how the AI was prompted, or what insight led to the search that found it.[1]

One mathematician described it this way: the conjecture sounded intuitively very true, so people had been trying to prove it. Nobody expected a one-sentence counterexample. The result is surprising both in its appearance and its nature.[1]

The Two-Variable Question

The counterexample disproves the conjecture with three variables. A version with two variables could theoretically still be true. So the conjecture is not entirely dead, just mortally wounded. Mathematicians will now pick through what remains.[1]

What This Means for Math

There is a difference between finding counterexamples that disprove conjectures and building whole new branches of mathematics. Andrew Wiles solved Fermat's last theorem by creating a hundred pages of new mathematics, building an entire theory along the way. The interesting stuff in math is often not "we ticked off this conjecture" but the mathematics you have to build to get there.[1]

Still, AI has now proven it can do this. A researcher at a university in China said AI can already produce master's degrees in mathematics, and within a year will produce PhD-level work. The question is what comes next. For 87 years, the Jacobian conjecture was one of those problems that sat in the background, something everyone assumed was true but nobody could prove. Now, with 216 characters, it is gone.[3]

That is not a hundred pages of new theory. It is a one-line answer found with the help of a machine. Whether that represents progress or simply a shortcut depends on what mathematicians do with the gap that is left behind.[1]