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AI has been on a streak of solving problems that stumped mathematicians for generations, and now, Anthropic's Claude has added another to the list.
Only months after an OpenAI model cracked a decades-old conjecture from Paul Erdős, a mathematician has used Anthropic's Fable model to settle an 87-year-old question, and experts are calling it the hardest maths problem AI has helped solve so far.
Levent Alpöge, a mathematician at Harvard University, wrote on 19 July that the Jacobian conjecture is false, backing it up with a tiny counterexample just 216 characters long.
Roughly speaking, the conjecture proposed that a certain type of mathematical function would also work in reverse. It was first set out by Ott-Heinrich Keller in 1939 and later earned a place on Stephen Smale's 1998 list of 18 problems he considered the most formidable challenges for mathematicians of the 21st century.
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For decades, the effort went into proving it true. Alpöge did the opposite, and did it in a single line.
Abhishek Saha, of Queen Mary University of London, said AI's recent run in maths had already been startling, but this went further.
“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.”
However, exactly how Alpöge used Fable to reach the counterexample is still not fully clear.
“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 noted.
He added that the result was surprising not only because of the AI involvement, but because many mathematicians had expected the conjecture to be true. “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 explained.
For Chris Bowman-Scargill, of the University of York, the discovery also raises a broader question about what matters most in mathematics.
“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,” stated 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.”
