Anthropic announced that an as-yet-unreleased AI model made significant progress on the Riemann hypothesis, one of mathematics’ longest-standing unsolved problems about the distribution of prime numbers. The model increased the lower bound of solutions for which the hypothesis holds true, a result confirmed by two of Anthropic‘s in-house mathematicians and formalized using the open-source proof assistant Lean. The Riemann hypothesis has remained unsolved for over 150 years, with a $1 million bounty for a general proof still unclaimed. While contemporary AI models cannot solve it entirely, this result shows they can make more progress than expected, reopening questions about AI’s ability to discover new scientific and mathematical ideas.

The way the result was achieved is particularly noteworthy. An Anthropic staff member without significant mathematical training prompted the model to ‘take a real stab’ at proving the hypothesis and then left it to coordinate the task over the following day and a half. The model tested 650 different ideas, coordinating across 60 sub-agents and spending 31 million in total (likely compute tokens). Out of the 60 sub-agents, two were responsible for developing the key mathematical ideas, 13 contributed ideas, 30 attempted but failed to develop new ideas, 13 served as validators to check correctness, and the final two helped write the initial paper.

This result adds to a string of mathematical breakthroughs driven by large language models. Earlier this year, a number of Erdos problems were solved by AI models, and OpenAI released ten major results proved by its internal Astra model. A separate effort from Anthropic disproved the long-standing Jacobian conjecture. The growing body of results has caused both excitement and concern in the mathematical field. In a public declaration signed in June, a group of prominent mathematicians raised concerns that AI could undermine critical values of the field, particularly the standard that mathematical proofs should be attributable to specific authors who take credit and assume responsibility.

However, the field remains split on how mathematicians should approach these new techniques. Fields Medal winner Timothy Gowers questioned whether the influence of AI might change mathematics in a more positive and complex way. In a blog post responding to the declaration, Gowers wrote: ‘If we arrive at a world where mathematical theorems are no longer associated with mathematicians, maybe that won’t be any more problematic than the fact that stars aren’t named after astronomers and most aren’t named at all.’