🧮 AI helps mathematicians solve problems that have been unsolved for decades

• Since October, AI tools have helped move about 100 of Paul Erdős' mathematical problems into the "solved" category.
• Large language models function as powerful research assistants that can find and combine existing mathematical results in new ways.
• Eleven top mathematicians have challenged AI with unpublished proofs in the competition First Proof, and the results are now being reviewed.

A hundred problems solved since October

The legendary mathematician Paul Erdős left behind 1,179 unsolved mathematical conjectures. Since October last year, AI tools have helped move about 100 of them into the "solved" category, according to a compilation by mathematician Terence Tao.

It started when mathematician Mehtaab Sawhney at Columbia University fed one of the Erdős problems into ChatGPT. The model found a reference to an existing solution immediately. Together with colleague Mark Sellke, he then used ChatGPT to dig up forgotten solutions to nine other Erdős problems, plus partial solutions to eleven more.

The bulk of the AI's assistance has been a form of advanced literature search. But in many cases, the language models have combined existing theorems to create new or improved solutions. In at least two cases, a language model constructed an entirely new and valid proof with minimal human input.

More than a search engine

Google's Gemini found a remark buried deep in a paper from 1981 that unknowingly solved Erdős problem number 1089. But the capabilities of language models extend beyond pure literature search.

Andrew Sutherland, a mathematician at the Massachusetts Institute of Technology, describes the language models as useful research assistants. He believes that mathematicians whose only experience with the models is older versions do not yet understand how capable they have become. Sutherland himself has had interactions where a model pointed him toward a result that allowed him to prove something he was stuck on.

The First Proof competition

Eleven top mathematicians have now launched First Proof, a new test of AI's mathematical abilities. They selected discrete chunks of proofs they had completed but not yet published and posed these as a challenge to AI. The problems cover a wide range of areas and vary in complexity. According to Daniel Litt, a mathematician at the University of Toronto, a system that could solve all the problems would be very useful for professional mathematicians.

The language models were given one week to produce proofs for the ten problems. The time limit was shorter than the time it took the team's own mathematicians to solve their respective problems.

By Monday, the team's emails and social media pages were inundated with claimed solutions. A Discord server hosting discussions about the challenge quickly gathered hundreds of members.

Verification is a challenge

Familiar problems quickly arose. First Proof was intended to go beyond pure literature search, and the team tested its questions on language models to ensure no answers existed in training data. But an online solution still surfaced to a problem from Fields Medal winner Martin Hairer, who had overlooked a partial proof on his own website archived by the Wayback Machine.

Verifying submitted solutions is resource-intensive. The models produce answers that sound convincing in about 90 percent of cases, but Daniel Litt has reviewed many of the circulating proofs and found them to be largely incorrect. A few, however, may be correct.

Mathematicians move to tech companies

In January, Ravi Vakil, current president of the American Mathematical Society, published a preprint together with two other mathematicians and two researchers from Google. They documented how Google's language model helped them reach a proof.

Several mathematicians predict that 2026 will be the year when results with AI as a stated contributor first pass peer review in major mathematics journals. Sawhney has taken academic leave from Columbia to work for OpenAI. Carlo Pagano, who collaborated with Google's DeepMind team on several Erdős problems, has started a position at Google DeepMind.

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