AI is transforming mathematics. Here’s where its limits still lie

Mathematics has a long history of such bold claims eventually being overturned. There are a great many examples of very hardened problems, some centuries old, giving in to human tenacity.

A conjecture made in 1782 by Leonhard Euler, one of the all-time great mathematicians, stood unchallenged for nearly 180 years until R C Bose, S S Shrikhande and E T Parker disproved it, earning themselves the nickname “Euler’s Spoilers” in a prominent New York Times article.

Fermat’s Last Theorem, proposed without proof in 1637, took 350 years to prove, finally falling in 1994-95. The Riemann Hypothesis, on the other hand, has resisted resolution for over 160 years and remains one of the seven Millennium Prize Problems — solve it and the Clay Institute owes you a million dollars

Why do mathematicians keep making these bets? Because the search itself tends to pay off even when the original question doesn’t get solved. George Boole’s 19th-century proposal for a symbolic calculus — now called Boolean Algebra — went on to become the backbone of computer science. Making bold assertions and waiting decades, or centuries, to see themselves vindicated or demolished is simply how the field works.

AI enters the arena

So, the Jacobian conjecture’s fate is interesting on its own terms. But the more striking story may be the growing role AI models — particularly Large Language Models (LLMs) — are playing in advanced mathematics.

The Jacobian result wasn’t a one-off. In May 2026, OpenAI announced that one of its internal reasoning models had produced a counterexample to a 1946 conjecture by Paul Erdős concerning unit distances between points in the plane — a problem that had resisted specialists for 80 years. The construction put forward by AI was subsequently verified and written up by nine outside mathematicians, including Fields medalist Tim Gowers. Will Sawin of Princeton went on to sharpen the result further.

Three years ago, when ChatGPT was new, I suspect many mathematicians — myself included — assumed LLMs were good for retrieving information already sitting in documents, but incapable of original mathematical thinking. That assumption has not aged well. At the 2024 International Mathematical Olympiad (IMO), Google DeepMind’s AlphaProof and AlphaGeometry 2 — specialised theorem-proving systems, not general LLMs — solved four of six problems for a silver-medal score of 28/42. By IMO 2025, Google’s Gemini Deep Think worked entirely in natural language, was officially graded by IMO coordinators under contest conditions and solved five of six problems (35/42) — a gold-medal score. OpenAI’s own model reached an identical 35/42 the same year, graded independently by three former IMO medalists.

At IMO 2026 in Shanghai, OpenAI’s GPT-5.6 reportedly solved all six problems on a single unsteered attempt. Independent testing by researchers including SignalPilot Labs suggests several frontier models — GPT-5.6 Sol, Anthropic’s Claude Fable 5, and others — reached a perfect 42/42 score, with Fable reportedly the fastest to finish. These results are self-reported or independently reproduced rather than certified by IMO coordinators. Even with that caveat, the trajectory from silver to gold to a plausible perfect score in three years is a genuine step change from what reasoning models could do at the start of this decade.

Solving critical problems

Much of mathematical research begins with a question — either posed by another mathematician or arising from a real-world application. Often the real challenge is figuring out under what assumptions the question has an answer. If a theorem is already known under stronger assumptions, a great deal of research consists of finding which of those assumptions are truly essential and which can be weakened or dropped.

This typically means exploring many possibilities, testing examples, and hunting for patterns. This is exactly the kind of work AI excels at. It can rapidly sift through a vast space of possibilities, surface promising directions, and — when guided by well-formulated prompts and given the relevant mathematical context — even generate proofs. For problems of this type, AI can dramatically accelerate research.

But not every mathematical breakthrough follows this pattern. Some of the deepest advances come from entirely new concepts, unexpected insights or radically different ways of framing a problem — not from searching more efficiently within an existing framework.

Whether AI can reliably produce that kind of conceptual leap is still an open question. So the honest answer is: yes, AI can already help solve many important problems in mathematics, and it will likely transform how mathematical research is done. But it would be premature to conclude that AI can solve every mathematical problem.

The catch

Here is the crucial limitation: an LLM has no built-in guarantee that what it says is true. It generates text that is statistically plausible, not necessarily correct — it is built to produce the most probable answer to your prompt, not a certified one.

There’s a pleasing echo of this, two centuries early, in Boole himself. Soon after proposing his symbolic calculus, he wrote a book with the sprawling title An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities (1854). Boole was already trying to formalise reasoning as something you could compute — logic on one hand, probability on the other. Today’s LLMs are, in a sense, living proof of the second half of that title: engines of probability rather than certainty.

But users must remember that LLMs have no innate fact-checker and no direct access to the world beyond its training. It can occasionally produce convincing but entirely wrong statements — the phenomenon now widely called hallucination. It is built to generate the most probable continuation of a conversation, not to certify the truth of every sentence. That combination — extraordinary linguistic and mathematical fluency alongside occasional factual unreliability — defines both the power and the limits of today’s LLMs.

The century-old idea underpinning large language models

A large language model (LLM), like ChatGPT or Claude, doesn’t ‘memorise’ the vast amounts of content it is trained on. Rather, it builds an intricate statistical map of language by observing how words and ideas co-occur across countless contexts.

But the idea of treating language statistically goes back at least to 1913, when the mathematician Andrey Markov analysed the sequence of vowels and consonants in Alexander Pushkin’s novel-in-verse ‘Eugene Onegin’, effectively inventing what is now called the Markov chain.

That thread — modelling language as a chain of probabilities — ran through a century of research until 2017, when eight researchers at Google Brain published a paper with the deceptively modest title ‘Attention Is All You Need’. It introduced the Transformer, the architecture underlying essentially every major LLM since, including ChatGPT.

A century of statistical groundwork, years of relentless engineering and one architectural breakthrough brought language modelling to a place none of us, staring at early search engines, would have believed possible.