AI & Models
Anthropic says an unreleased model advanced a more than 150-year-old math problem
Anthropic said Monday that an unreleased AI model made significant progress on the more than 150-year-old Riemann hypothesis, working largely on its own for a day and a half.
The Riemann hypothesis, a 150-year-old mystery about how prime numbers are distributed, remains one of mathematics’ most famous unsolved problems — carrying a $1 million bounty for a full proof that has gone unclaimed. No AI model has solved it either, but Anthropic says its systems can make far more progress on it than expected, a finding likely to revive debate over whether AI can genuinely help discover new science and mathematics.
On Monday, Anthropic announced that an as-yet-unreleased model had made significant progress on the hypothesis, significantly increasing the lower bound of solutions for which it holds true. The way it happened is notable: an Anthropic staff member without significant mathematical training prompted the model to “take a real stab” at a proof, then left it to coordinate the work largely on its own over the following day and a half. In total, the model tested 650 different ideas for solving the problem, coordinating across 60 subagents and spending 31 million output tokens, with two of those subagents credited with developing the key mathematical ideas and 13 more serving as validators. Two of Anthropic’s in-house mathematicians confirmed the finding, which was then formalized using the open source proof assistant Lean.
The result adds to a string of AI-driven mathematical breakthroughs this year. A number of Erdos problems have been solved by AI models, with the release of more powerful models leading to more impressive results. OpenAI recently released a set of 10 major results proved by its internal “Astra” model, while a separate effort from Anthropic disproved the longstanding 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 a core value of the field: that true proofs be “attributable to specific authors who take credit for their discovery and assume responsibility for their correctness.” The field remains split. Responding in a blog post, Fields Medal winner Timothy Gowers questioned whether AI’s influence might change mathematics in a more complex and positive way, writing that a world where theorems are “no longer associated with mathematicians” might be no more problematic “than the fact that stars aren’t named after astronomers and most aren’t named at all.”
Why it matters
If an unreleased model is already pushing on a century-old open problem largely unsupervised, the debate over how mathematics credits and verifies discovery is likely to intensify well before such models are publicly available.