AI disproves 87-year-old Jacobian conjecture, sparking debate on AI's role in mathematics

AI's solution to 87-year-old riddle takes mathematicians by surprise | New Scientist

A mathematician has cracked an 87-year-old conundrum with the help of AI and announced the solution unceremoniously in a tweet. The finding is the most difficult mathematical problem yet solved by AI, say experts. Levent Alpöge at Harvard University wrote on X on 19 July that the Jacobian conjecture - which academics have spent decades trying to prove was true - is actually false, giving a tiny, 216-character counterexample as proof. The Jacobian conjecture - which suggests that a certain type of mathematical function would also work in reverse - was formally set out by Ott-Heinrich Keller in 1939. It was also on an influential list of 18 fiendishly difficult problems for mathematicians to tackle in the 21st century drawn up by Stephen Smale in 1998. Alpöge did not respond to New Scientist's request for interview, but said in his tweet that part of the work was down to his "close friend fable"- seemingly referring to AI company Anthropic's Claude Fable 5. Alpöge thanked Fable for working during the World Cup final. Anthropic did not respond to a request for comment. Abhishek Saha at Queen Mary University of London says AI's recent advances in mathematics, such as the OpenAI model that recently cracked a decades-old conjecture by Paul Erdős, have been surprising, but this latest finding has stepped things up significantly. "Probably this is the biggest conjecture that AI has played a significant role [in proving or disproving] so far in mathematics," he says. "This is a pretty big deal. AI has [made] remarkable progress in the last year." The single line of mathematics posted by Alpöge was simple to verify and many mathematicians have already done so, says Saha. Now the big question is how it was done. "There are some problems that are very hard to solve but once a solution is there, they are relatively easy to check. So this is like that," he says. "I don't know how he did it, what exactly was the prompt to give Fable, because if one were to search everything, it wouldn't quite work, so obviously there was some insight also which is not currently published." Saha says both the appearance and the nature of the result is a surprise. "People have been trying to prove it [the Jacobian conjecture] because it sounds, intuitively, very true. I don't think that many people have been trying to disprove it. And now we have this one-sentence counterexample," he says. There are still open questions, says Saha. For example, this new counterexample disproves the conjecture with three variables, but a version with two variables could theoretically still be true. Chris Bowman-Scargill at the University of York, UK, says mathematicians have in some ways already adjusted to the shocking new capabilities of AI, but there is a difference between finding counterexamples that disprove conjectures and building whole new branches of mathematics, which still requires human creativity. "If you look at Fermat's last theorem [which was solved by Andrew Wiles in 1994], you had to create a hundred pages of new mathematics - you had to build a whole big theory in order to solve a conjecture," says Bowman-Scargill. "And often the interesting stuff in maths isn't 'oh, we've ticked off this conjecture, yay', it's more the stuff you have to build along the way in order to solve the conjecture." "I think [AI] has sort of proven that it can do this, so you're now like, OK, what next?," he says. Ivan Fesenko at Westlake University in China believes what comes next is increasingly capable AI models that will solve ever more complex problems, disrupting the field as they go. "Right now, AI can already produce master's degrees in mathematics. In one year, they will produce PhD degrees in mathematics. And then the question arises, do we really need so many mathematicians around if AI can do such things so nicely?" says Fesenko. "So basically we're talking about fundamental change in mathematics."

A mathematician used Fable 5 to disprove a major math problem

While the rest of the world was watching the World Cup on Sunday evening, mathematician Levent Alpöge casually announced on X that he had used Anthropic's Fable 5 to disprove the Jacobian conjecture. The Jacobian conjecture is a long-standing open problem in algebraic geometry that's bedeviled highly accomplished mathematicians for almost 90 years. It was included in "Smale's problems," a list of unresolved math problems put forward by mathematician Stephen Smale in 1998. But that proved no obstacle for Fable 5, Anthropic's latest frontier AI model. (You may remember Fable 5 as the public version of Claude Mythos Preview, the AI model that Anthropic says has such advanced cybersecurity capabilities that it was too dangerous for public release.) Alpöge is part of Harvard's Society of Fellows, and his Linkedin profile notes an affiliation with Anthropic. We've reached out to Anthropic with questions about Alpöge's work. Meanwhile, for those of us who don't understand polynomial functions from n-dimensional space, the question is: Does this represent a major breakthrough in artificial intelligence, or mathematics, or neither? "This did not cause me to update my priors about what AI can and can't do," professor and mathematician Andrew Blumberg told Mashable. "This is exactly the kind of thing I would expect AI to be able to do. If there was a counterexample that was concise and easy to state that people haven't found because it's a pain to search through all this stuff, AI will find it." Blumberg, who has a joint appointment in mathematics and computer science at Columbia University, is involved in the First Proof project, an effort to test the capabilities of frontier large language models in solving research-level mathematics. So, it's safe to say he understands mathematics and AI better than most of us. Providing a counterexample to the Jacobian conjecture is still a significant achievement for AI in mathematics, Blumberg notes. But he also says there's a big difference between providing a positive proof for a major math problem and providing a single counterexample that disproves it. Here's the metaphor Blumberg used to explain. "Suppose that Moses came down from the mountain with tablets, and on the tablet was written, 'Cancer can be cured.' Would you care? You don't just want the answer to the question. You want to learn something from the answer. And the reason Smale thought this problem was important is because he thought that if we solved it, we would understand more things about the about the way nature is structured." In other words, Blumberg says, "this counterexample tells us essentially nothing. It's just, you know, there are a lot of polynomials, and it's hard for people to check them all, but it's not hard for machines." The Jacobian conjecture isn't the only math problem AI has tackled in recent months. OpenAI announced in May an "internal model" had disproved the Erdős unit distance conjecture, a central conjecture in discrete geometry. However, in that case, the result was more productive, Blumberg says. That's because OpenAI didn't just provide a counterexample -- but a disproof of the problem itself. "The disproof there already has led to interesting things because experts in the area unpacked what was going on in the disproof and then used it to do other things," Blumberg says. And that's not necessarily the case with the Jacobian conjecture counterexample, he adds -- while conceding he's not expert enough to see if there's something special about the structure of the counterexample that we can learn from. "But it's not in and of itself interesting." Disclosure: Ziff Davis, Mashable's parent company, in April 2025 filed a lawsuit against OpenAI, alleging it infringed Ziff Davis copyrights in training and operating its AI systems.