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On July 20, mathematician Levent Alpöge, who works at the artificial intelligence company (AI) Anthropic, dropped a casual tweet on X. He claimed the company’s advanced large language model Claude Fable 5 had produced a counterexample to an 87-year-old open problem in algebraic geometry called the Jacobian conjecture. By working on it during the FIFA World Cup final, AI had helped disprove a conjecture that many mathematicians had been working on for decades — and the world got the news about it in a short social media post.
This was not AI’s first major mathematical conquest. On May 20, OpenAI released a series of tweets on X claiming that its internal, general reasoning model had found a solution to the planar unit distance problem, an open problem proposed by Hungarian mathematician Paul Erdős in 1946. Many expert mathematicians who verified the solution realised that if humans had solved this on their own, the result would be out in a top journal.
The floodgates were opened. On August 1, OpenAI announced that its newest model ‘Astra’ had solved ten more open maths problems, three of which were from the legendary list of Erdős problems that included the unit distance conjecture. AI had been solving other less exciting Erdős problems and International Math Olympiad questions even before tackling the unit distance problem — but this latest set had many people sit up and take notice. As Columbia University theoretical computer scientist Henry Yuen tweeted, “I didn’t just hear about these problems from my friends or from seminars. I feel their importance in my bones; I deeply care about the answers to these questions.”
In the most significant advance yet, on September 8, OpenAI announced that an unreleased internal model had solved a Millenium Prize Problem called the Navier-Stokes existence and smoothness problem — one of six heavily researched maths problems with a million-dollar bounty.
Spurred by rumours that their rival Anthropic might be closing in on a solution, OpenAI went all guns blazing, deploying 10,000 autonomous AI agents and millions of dollars’ worth of computing power and found a solution in 88 hours.
But even as it rattles the maths world, the achievement is already shrouded in controversy. On September 7, New York University mathematician Tristan Buckmaster revealed he and Alpöge had been personally building on previous work by other mathematicians and working on a solution for a simplified version of the problem using similar methods for about a year, and they were assisted by both Anthropic and OpenAI models in their efforts.
While OpenAI has denied specifically accessing the duo’s data, the company noted “they cannot rule out that de-identified data from [the duo’s] usage of their products helped improve their models.”
This flurry of AI-generated proofs has left mathematicians in various emotional states. Some are cautiously excited about how the fast-improving models can help them in their research. Some are critical of the AI companies’ commercial motives behind their relentless mining of open maths problems and are insisting on more careful regulation. And some are left asking existential questions: if AI can solve important problems at this alarming rate, what role will mathematicians play in the future? Will they have jobs in five or ten years? And what if AI generates a complicated proof of a major problem that no one can understand?
The rise of AI has generated an unprecedented level of discourse and introspection among mathematicians about what they really value and enjoy about doing maths, with one sentiment prevailing among them. As MIT mathematician Andrew Sutherland put it: “I think it is going to change the way mathematicians do research going forward.”
‘Like a grad student army’
Andrew Booker, a number theorist in the University of Bristol, had been stuck solving a central problem in number theory with a collaborator for about ten years. Called the converse problem for L-functions, they would work on it for a few weeks, give up out of frustration, and periodically return to it. “And suddenly, that all changed in April,” he said.
Dr. Sutherland had gifted Dr. Booker a gift subscription to Claude Max, finally pushing him to use Claude Code, which is Claude’s AI coding agent, on the problem. He wasn’t expecting much — but the AI model wrote 30,000 lines of code and about 200 pages of maths in the first week, “an amount I would write in a year,” he said. Although they haven’t completely solved the problem yet, “within two weeks we had made more progress than in the previous five years.”
The frontier AI models released in the last year have become a lot better at reasoning compared to the older, more sycophantic versions. “It works almost eerily like a human,” said Dr. Booker. “It reasons things out in the same way. It makes the same mistakes as a human.” It needs expert guidance in the hard problems, but it can, on occasion, find something very obvious that humans completely missed.
“They’re not perfect, but they’re operating at the level of a grad student, with the difference that they’re a hundred times faster than grad students,” Dr. Booker said. “It’s like having a grad student army, if you like.”
That feels powerful because it has made thinking, which is usually difficult, much cheaper. Humans might have several ideas but cannot work on all of it to the same extent the way AI can. “And if it continues improving [at this] pace, it won’t be long before it matches the skill set of the best mathematicians in the world,” Dr. Booker said.
Dr. Sutherland got interested in using machine learning and AI in mathematics research in early 2022, before the large language models era. As a computational number theorist, he realised that AI/ML tools could help find interesting patterns in large amounts of data. Researchers slowly realised that the models were like a “more intelligent search engine”, he said, being able to sift through literature and find useful connections between vastly different fields of maths.
But a major turning point for him was when Claude’s Opus 4.5 came out late last year, which is an AI agent that can code, produce proofs, and formalise them — i.e. convert them to a computer science-based language (like Lean), which can be verified by a machine. In the last six months or so, the newest AI tools are “both good at writing code and very good at math, and that combination has been very, very valuable,” Dr. Sutherland said. “I’m still the one driving the process, but having these tools lets me go faster and see further than I could before. It’s an exciting time from that perspective.”
‘It writes like an alien’
Nevertheless, he understands why these powerful AI tools can be disconcerting, especially for theoretically inclined mathematicians who enjoy solving very hard, complicated problems. Instead of thinking deeply about a problem for many years, one can now get an answer from a model immediately with a few well-placed prompts, he said.
Some maths students are worried about whether they will have jobs in the future, and whether mathematicians will become obsolete if AI models become better than them at solving problems. One of the winners of the 2026 Fields Medal, Jacob Tsimerman, has stopped taking graduate students on because he is uncertain about whether the traditional mathematical career he trains them for now will even exist in a few years. (He has since moved to OpenAI to work on AI safety).
AI’s proofs are not perfect, however. The models tend to gloss over the hard parts of the proof, Dr. Sutherland said — which is usually the key, intuitive step that humans may struggle with and ultimately gain insight from if they tackled it on their own. Also, models’ proofs are incredibly hard for humans to understand: as it rapidly borrows concepts and vocabulary from vastly different mathematical fields and learns on the go, it tends to make up weird terminology as it’s working. It also does not cite all the previous research it is building on.
“It writes like an alien sometimes,” Dr. Booker said. “If left unchecked for too long, you may be left with text that’s completely incomprehensible. It’s [not] meaningless gibberish — the AI understands it — but it’s so far removed from standard terminology that I can no longer parse it.”
Dr. Booker eventually learnt that he must supervise the AI assistant to keep it grounded, prune its long arguments, and rewrite some of its proofs to add more intuition, to keep it less dry and more appealing to human readers. Researchers have called OpenAI’s proof write-ups “horrendous”on X, implying that the companies are not communicating their results in a way that others can gain insights from them.
There is, however, an unsettling way to circumvent mathematicians entirely: AI can rapidly generate and formalise its own long, incomprehensible proofs, which can then be machine-verified.
In an essay titled ‘Mathematics in the Age of AI’, the Australian-American mathematician and 2006 Fields Medal winner Terence Tao discussed how the field could shift from “proof scarcity to proof abundance”, and with more proofs than humans can deal with, this may also end up in “proof indigestion”.
“Something that’s happening with these large language models releasing these proofs one after another is we have not had time to chew and understand these proofs,” Tata Institute of Fundamental Research, Mumbai, computer scientist Prahladh Harsha said. A nightmare situation, Columbia University mathematician Michael Harris said, is that AI will generate a long, incomprehensible proof of the Reimann hypothesis — one of the most famous open problems in mathematics — and humans will just have to accept it without understanding it.
Training students differently
“It’s a very confusing time right at this moment, because nobody knows what the rules are,” said Dr. Booker. Many mathematicians are worried that the AI tools are developing at a rate faster than human institutions can adapt.
Dr. Sutherland, however, doesn’t think mathematicians will become obsolete. There is still value in humans being able to understand the proof on their own, to learn not just “what is true but why it’s true,” he said. With infinitely many mathematical problems and only a finite number of computers, he doesn’t think that mathematics will be “solved” by AI. But with AI trivialising the easier problems, “the bar to make a significant contribution is going to rise,” Dr. Booker said.
Students may need to be trained differently, as “maybe pure, problem-solving ability is not going to be as highly valued in the future as it was in the past,” Dr. Sutherland added. Improving the exposition of AI-generated proofs or using AI assistance to tackle and understand hard problems may become more important.
But people passionate and excited about maths should still pursue it, he said, willing to work with students who engage with AI tools in their research. Current students’ careers may look very different from their own advisors’, he noted, but that doesn’t mean they won’t have one. However, young students should not use AI to solve homework problems as that will interfere with the thinking and learning process in their early stages of their training, he advised.
Terence Tao teaching analytic prime number theory in January 2025.
| Photo Credit:
Natecation (CC BY-SA)
‘Generous with ideas’
After a 2025 conference titled ‘Mechanization and Mathematical Research’ at Leiden University, a small group of the participants stayed in touch and drafted a set of much-needed guidelines for responsible AI use. They came to be called the ‘Leiden Declaration on Artificial Intelligence and Mathematics’. With many distinguished endorsers, including Dr. Tao, the June 2026 declaration has emerged as an important step in understanding how AI tools affect mathematics, with recommendations on how to engage ethically with them.
Dr. Harris and University of California San Diego mathematician Karthik Ganapathy, two members of the working group involved in drafting the declaration, are both critical of how AI companies do not release a detailed account of how exactly their internal models have solved a problem — in most cases, the result is declared as a press release or as an unceremonious tweet.
“The best mathematicians are very generous with their ideas,” Dr. Ganapathy said, unlike giant AI corporations that are incentivised to not be open about their methods. This lack of transparency also makes it hard to ascertain how much the advanced mathematicians working at AI companies played a role in the proof.
Harvard University mathematician Nina Zubrilina agreed that the companies should ideally be releasing the prompts they used in the discovery process to get more insight into the model’s reasoning. “There’s a certain amount of intentional mysticism around how they arrive at certain results, which I find counterproductive for understanding something deeper,” she said.
Ideally, these solutions should lead to a deeper theory or “set the groundwork for something bigger.” Simply resolving a decades-old conjecture “is not really the right metric for success,” she added. “It makes for good headlines, but it’s not necessarily what is valued kind of the most for people who do mathematics for a living,” she said, while also noting that AI might improve at theory-building in the future.
In his essay, Dr. Tao insisted that only when a proof is written in a way that humans can understand and explain it, the broader community accepts it, and is ultimately included in textbooks or reference material that is taught to the next generation of students is the process of solving any problem complete. This last step of canonicalisation “requires broad, deliberative consensus, and it is the stage least amenable to optimisation by AI tools,’ he wrote. “It is also, in my view, the most valuable part of the entire process.”
OpenAI CEO Sam Altman speaks at the 2026 Infrastructure Summit, in Washington, D.C., U.S., March 11, 2026. On July 29, OpenAI announced it will allow one lakh scientists, mathematicians, and engineers to access its frontier models for free — but Indian, Russian, and Chinese institutes were excluded from this list.
| Photo Credit:
REUTERS
Maths as cultural capital
Although he may use AI for routine work, Dr. Ganapathy wants to think on his own to preserve the pleasure of doing pure maths. “It’s not like we want to completely offload all aspects of [doing mathematics] to a machine,” he said, saying how no mathematician is “asking OpenAI to build a model which can reason this way.” Pure maths problems might eventually lead to applications only if humans are able to think together and apply the insights they learn in innovative ways, and AI using opaque methods to produce impenetrable proofs will hinder this process.
“Corporations have no material interest in abstract mathematical research,” said Dr. Harris, who thinks there are two reasons driving the AI companies: they think rigorous mathematical reasoning may bring the models closer to artificial general intelligence — a vague, hypothetical intelligence that will match and/or beat humans in all cognitive tasks; and demonstrable mathematical prowess may make investors happy. Academics’ “goals are trying to understand the math [and] science behind these [proofs], but [the companies’] goals are to show that they have built very smart models and monetise that,” Dr. Harsha said.
Also, not everyone can access the frontier AI models: students in India don’t have access to them the way US students do, Dr. Harsha said. On July 29, OpenAI announced that it will allow 100,000 scientists, mathematicians, and engineers to access its frontier models for free — but Indian, Russian, and Chinese institutes were excluded from this list, most likely because of geopolitical tensions.
India, with no frontier model of its own, unlike the U.S. or China, might struggle to keep pace with theoretical scientific advances at this rate, even though the large country had previously enjoyed an edge over others in the theoretical sciences. “Mathematics is certainly one way that [AI companies are] trying to gain cultural capital, which will inevitably help them change policy,” said Dr. Ganapathy.
Dr. Harris thinks that instead of treating AI as an inevitable natural phenomenon, mathematicians must recognise it as a product of giant, commercially driven corporations. Are they then comfortable with these corporations dictating the direction of mathematical research?
“The interests of mathematics are protected by mathematicians and not by the corporations,” Dr. Harris said. “As long as the mathematicians are free from external interference or attempts to shift the funding strategies in the direction more beneficial to the investors, I think they can be trusted to use the AI technology in a way that is in general interest.”
Rohini Subrahmanyam is a freelance journalist in Bengaluru.
