AI Cracked Ten Hard Maths Problems in One Summer. Now Mathematicians Are Fighting Back.
OpenAI's models solved open problems that stumped humans for decades. Terence Tao says mathematicians have until the end of 2025 to decide who controls their field's future.
Key points
- In May 2026, an OpenAI model disproved a conjecture by Paul Erdős that had stood since 1946.
- On 1 August 2026, OpenAI announced solutions to ten further open problems in mathematics, including three more Erdős conjectures.
- A group of mathematicians published the Leiden Declaration, calling for disclosure rules and ethical guidelines for AI use in research.
- Fields Medal winner Terence Tao warned that mathematics must reorganise urgently or cede control to technology companies.
- AI-generated proofs are sometimes technically correct but nearly unreadable, and critics say the tools routinely fail to credit the human work they draw on.
This past summer, an AI model worked through some of the hardest unsolved problems in mathematics and solved them, one after another, in ways that surprised even the researchers running the experiments.
May 2026 was when OpenAI announced that one of its models had disproved a conjecture, a mathematical idea believed to be true but not yet proved, that the prolific Hungarian mathematician Paul Erdős had proposed in 1946. Scatter dots across a page: Erdős believed the arrangement producing the greatest number of equally spaced pairs was a neat, regular grid. The AI found that messier, less symmetrical arrangements can beat the grid by a wide margin, using principles drawn from advanced number theory.
Then, on 1 August 2026, OpenAI published results on ten more problems, covering territory from sphere packing, which asks how efficiently you can stack identical balls in many dimensions, to post-quantum cryptography, the branch of security designed to resist computers that do not yet exist. Three of the ten were problems Erdős himself had posed. Our 11 August report covered the credit dispute that followed one of them, finding it a warning about how AI labs talk about the people whose work they build on.
What did the AI actually solve?
The list spans several areas of pure mathematics, each considered difficult by professional researchers.
| Problem area | What was claimed |
|---|---|
| Sphere packing | New upper limits on how densely spheres can fit in high dimensions |
| Binary and spherical codes | Sharply improved bounds on error-correcting codes used in communications |
| Non-sofic groups | First confirmed existence of a class of mathematical structures long debated |
| Connes's rigidity conjecture | Disproof of a decades-old conjecture in algebra |
| Multicolor Ramsey numbers | Resolved Erdős problem 183 with a superexponential lower bound |
| Closest vector problem | New hardness result relevant to post-quantum cryptography |
These are not variations on a single theme. Different branches, different languages, different tools.
Should mathematicians be worried?
Many are. Signed by a growing group of researchers, the Leiden Declaration on Artificial Intelligence and Mathematics calls for new professional standards before AI reshapes the field on someone else's terms. Among its concerns: systems sometimes generate plausible-looking proofs containing hidden errors, and software trained on decades of published mathematics often fails to credit the human work it draws on.
For individual researchers, the declaration's asks are concrete: disclose when you use AI tools, take personal responsibility for checking correctness, and push back when private companies try to define what counts as good mathematics.
Terence Tao, a Fields Medal winner widely regarded as the leading mathematician alive today, put the urgency plainly in a New Scientist interview flagged on Hacker News. "We don't have to passively accept changes by external forces," he said. "We can't just passively prove our theorems; we have to organise, become activists, get a little political."
His specific complaint about AI-generated proofs is worth hearing. A model will spend pages on something any professional considers obvious, then compress the genuinely hard step into two sentences. The reasoning is also opaque: companies do not publish their methods in full, and researchers do not yet understand how the models reach their answers.
Without action from the mathematics community itself, Tao warns, technology companies may shift the field's focus toward grinding out as many provable results as possible, regardless of whether those results matter. "They tried to redefine what our profession is," he said. "Mathematicians have more influence than they think."
What happens next?
Tao wants a significant cultural overhaul, and he wants it done by the end of this year. That is a short window for a field not known for moving quickly.
Practically, the steps are modest but real: say when you used an AI tool, check every proof before putting your name on it, and speak publicly about what mathematics is actually for. The Leiden Declaration frames these not as restrictions but as the minimum needed to keep the discipline honest.
The beat worth watching is attribution. Speed of proof generation gets the headlines, but who gets credit for the underlying mathematics, and whether AI labs acknowledge it, will shape whether human researchers stay invested in the field at all.
