Anthropic's unreleased model makes progress on Riemann hypothesis

On Monday, Anthropic announced a breakthrough that challenges longstanding assumptions about AI's mathematical capabilities. An unreleased Anthropic model made significant progress on the Riemann hypothesis—one of mathematics' most enduring unsolved problems, which has resisted proof for more than 150 years and currently carries a $1 million unclaimed bounty. The hypothesis concerns the distribution of prime numbers and represents a frontier of mathematical understanding.

What distinguishes this result is both its scale and its method. An Anthropic staff member without significant mathematical training initiated the effort by simply prompting the model to "take a real stab" at proving the hypothesis, then allowed it to work autonomously. Over roughly a day and a half, the model tested 650 different ideas, coordinating across 60 subagents—specialized sub-instances of itself—and consumed 31 million output tokens in total. According to a footnote in Anthropic's paper, the work was distributed across these subagents with specific roles: two developed the key mathematical ideas, 13 contributed supporting ideas to those agents, 30 attempted but failed to generate new ideas, 13 served as validators to verify the correctness of arguments, and two helped write the initial paper. The resulting progress—a significantly increased lower bound for solutions satisfying the hypothesis—was confirmed by two of Anthropic's in-house mathematicians and formalized using Lean, an open source proof assistant.

This is not an isolated achievement. A number of Erdos problems have been solved by AI models over the course of this year. OpenAI recently released a set of 10 major results proved by its internal "Astra" model, and Anthropic has separately disproved the longstanding Jacobian conjecture. The cumulative effect of these results has begun to reshape how the mathematical community thinks about AI's potential.

However, the progress has sparked considerable debate. In a public declaration signed in June, a group of prominent mathematicians raised concerns that AI could fundamentally undermine one of mathematics' core values: the standard that true mathematical proofs should be "attributable to specific authors who take credit for their discovery and assume responsibility for their correctness." Yet the field is not monolithic in its response. Fields Medal winner Timothy Gowers, writing in response to the declaration, questioned whether the influence of AI might ultimately change mathematics in a more complex and positive way. "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," Gowers wrote, suggesting that the loss of individual attribution may not constitute a genuine loss for the discipline.