On Sunday afternoon, while the rest of the world stared at the World Cup final, an AI model solved a problem that has plagued mathematicians since 1939.
When Kevin Buzzard woke up the next morning in London, the result was confirmed. At lunchtime that was all his fellow students in the pure mathematics department at Imperial College London could talk about; At the time of writing, Anthropic employee Levant Alpöge’s post announcing the result has more than 20 million views on X.
“It’s a big day,” Buzzard said Assets. “I personally think it’s a great time to be alive.”
It was the latest in a series of AI-driven mathematical breakthroughs. AI’s progress (or attack) on unsolved mathematics has increased rapidly since mid-2025, when models first solved five of six problems at the International Mathematical Olympiad. From there, the list of fallen problems quickly grew: the OpenAI model disproved an 80-year-old Erdős conjecture on combinatorial geometry in May, and in June, 16 researchers from 15 universities published the Leiden Declaration on Artificial Intelligence and Mathematics, calling on the profession to establish guidelines for transparency, attribution, and peer review before AI even redefines the meaning of mathematical knowledge.
Relegated to the role of shepherd, mathematicians must watch as AI solves these questions one by one, going into places the human mind cannot. Her reaction is a now familiar mix of fear and astonishment.
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An 87 year old problem
The problem is called the Jacobi conjecture and has been based on the work of the German mathematician Ott-Heinrich Keller since 1939. Essentially, it’s about what mathematicians call “maps” and the conditions under which one can determine the input based on a set of outputs. Since it was mathematics, it was based on another German work a century earlier: Carl Gustav Jacob Jacobi’s Jacobi Determinant. The main problem for modern practitioners is that until now they have been unable to prove Keller’s conjecture to be true or to find a reason why it was false.
On Sunday. Alpöge’s result satisfies the Jacobian determinant at every point in space – the determinant remains constant at −2 everywhere – and yet sends three different starting points to the same destination. That means it didn’t pass the test.
It’s a “very exciting” result, Buzzard said, one that shows the potential for language models to eventually achieve the “super mathematician” that Google deep learning scientist Christian Szegedy warned about about half a decade ago.
But it also leaves mathematicians wanting. The problem with current AI models solving pure mathematics is understanding the “how” without the “why,” explained Akhil Mathew, a University of Chicago mathematician who Alpöge credits with suggesting the problem to him. “You can check if it’s right,” Mathew said Assets“But it would be nice to be able to tell a story.”
Alpöge didn’t answer Fortune’s Please comment.
Why we have pure mathematics at all
Mathematicians have dealt with automation before. Most people with high school level math skills view the job as a calculation that computers conquered decades ago. “Then you go to college and when you take some advanced math courses, you learn that math is really just about logical thinking,” Buzzard said.
A calculator multiplies four-digit numbers faster than any human. What a mathematician adds is why: When Buzzard is told that 131 times 137 is four million, he doesn’t need to reach for the calculator – he knows that two odd numbers can’t make an even number. Understanding something, he said, means “integrating it into the brain so well” that you can generate the result from the idea itself.
Proving your knowledge of formal mathematics is a “proof,” a chain of logical steps that follow each other and end with the claim you made. Proofs are both the way mathematicians build and how they are measured. A good proof can run hundreds of pages and take months to explain to experts you can trust.
So far, AI does not yet have the capabilities to create such proof, Buzzard said. Creating a sophisticated 150-page proof requires hundreds of steps, and language models have a habit of filling in gaps with plausible-sounding filler. Because unlike a human colleague, the model does not risk reputation if she is wrong.
If this bottleneck breaks, however, it will be Buzzard’s own fault. His career project is Lean, a popular computer language in which proofs are checked by machines rather than by exhausted graduate students. He said the evidence had already been checked in Lean when he woke up. The moment proof-writing models meet his proof-checking machine, one of man’s last advantages in mathematics disappears.
The question of “taste”
Mathew was more tempered in his excitement, calling the moment “a very rapid and very worrying change…particularly for young mathematicians.”
Michael Harris, a professor of mathematics at Columbia University, wrote in a June paper in: Boston Review that the AI industry views reasoning or understanding as commercially worthless and human mathematicians as a “beta version of intelligence.” Yet mathematics, he argued, is one of the last examples of alienated labor, a field that people enter, in the words of Abel Prize winner Pierre Deligne, because one can make a living “by playing”—what Mathew calls “telling a story.” Even when Deep Blue “solved” chess in 1997 by defeating Garry Kasparov, people didn’t stop playing chess; they learned from it.
But perhaps subsidizing mathematicians to play sounds uninspiring to the public. Even before AI threatened their work, federal funding for mathematics research fell by about 72% due to Trump administration cuts to the National Science Foundation. Graduate student admissions to top research universities fell 15% this fall, the second straight year of decline; No sponsored students at all are accepted for the mathematics doctorate at George Washington University.
Some think that the death of mathematical professionalization is a good thing, that the machines are democratizing the entire enterprise, “gaming.” Garry Tan, president of Y Combinator, responded to the news on But Alpöge is no tinkerer; He’s a Harvard valedictorian who has spent a decade using algorithms to calculate exactly this kind of problem.
And that could be the trick to keeping people in the math circle, Buzzard said. Beyond calculation, even beyond reasoning, “understanding” essentially means knowing what to ask, what Silicon Valley has called “taste.”
“Humans have tried to get machines to ask questions, but they are disastrous,” he said. “All the questions they ask are either boring or obviously true or obviously false.” The field’s monuments — the Riemann hypothesis or Keller’s Jacobi hypothesis — are named for the people who erected them, not the people who settled them, Buzzard pointed out. “It’s not a coincidence. You have to be a brilliant mathematician to ask the right question.”
https://fortune.com/2026/07/21/ai-solves-jacobian-conjecture-levant-alpoge-claude-fable-5/
