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. The world's most celebrated mathematician says the field has until the end of 2025 to decide who controls its future.
Key points
- In May 2026, an OpenAI model disproved a conjecture by legendary mathematician 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, widely considered the world's greatest living mathematician, warned that the field must reorganise itself 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 sat down with some of the hardest unsolved problems in mathematics. It did not merely hint at solutions. It solved them, one after another, in ways that surprised even the researchers running the experiments.
The run started in May 2026, 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. The problem asked: if you scatter dots across a page, what arrangement produces the greatest number of pairs of dots that are exactly the same distance apart? Erdős believed the answer was a neat, regular grid. The AI found that messier, less symmetrical arrangements can beat the grid by a wide margin.
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.
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 |
The breadth is striking. These are not variations on a single theme. They come from different branches of mathematics, each with its own language and tools.
Should mathematicians be worried?
Many are. The Leiden Declaration on Artificial Intelligence and Mathematics, signed by a growing group of researchers, calls for new professional standards before AI reshapes the field on someone else's terms. The declaration warns that AI systems sometimes generate plausible-looking proofs that contain hidden errors, and that software trained on decades of published mathematics often fails to credit the human work it draws on.
The declaration asks individual researchers to disclose when they 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 recent New Scientist interview, first flagged on Hacker News: AI. "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."
Tao's specific complaint about AI-generated proofs is worth hearing. The systems, he says, will spend pages on something any professional would consider obvious, then rush through the genuinely hard step in a sentence or two. The reasoning is also opaque: the companies running these models do not publish their methods in full, and researchers do not yet fully understand how the models reach their answers.
His warning is direct. Without action from the mathematics community itself, 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," Tao said. "Mathematicians have more influence than they think."
What happens next?
Tao says the community needs 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.
For working researchers, the practical steps are modest but real: say when you used an AI tool, check every proof yourself before putting your name on it, and speak up 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.
