Anthropic’s unreleased model advances progress on Riemann hypothesis with multi-agent workflow

An unreleased Anthropic model made progress on the Riemann hypothesis, a 150-year-old unsolved problem in mathematics, by expanding the lower bound of solutions for which the hypothesis holds true. The advance was achieved without a formal mathematical background from the prompting Anthropic staff member, who instructed the model to "take a real stab" at proving the hypothesis. The model then coordinated the effort across 60 subagents over the course of a day and a half, testing 650 different ideas and spending 31 million output tokens in total.

Two of the 60 subagents were responsible for developing the key mathematical ideas, while 13 contributed ideas to these agents. Another 30 subagents attempted but failed to develop new ideas, and 13 served as validators to check the correctness of the arguments. The final two subagents assisted in writing the initial paper. The findings were confirmed by two of Anthropic’s in-house mathematicians and formalized using the open-source proof assistant Lean.

This development follows a string of mathematical breakthroughs attributed to large language models, including the solution of multiple Erdos problems and the disproof of the Jacobian conjecture. OpenAI recently published a set of 10 major results proved by its internal "Astra" model, highlighting a growing trend of AI-assisted mathematical research.

The increasing role of AI in mathematics has sparked debate within the field. In June, a group of prominent mathematicians signed a public declaration expressing concerns that AI could undermine core values of the discipline, particularly the principle that proofs should be attributable to specific authors who assume responsibility for their correctness. Fields Medal winner Timothy Gowers responded to these concerns in a blog post, suggesting that AI’s influence might reshape mathematics in complex and potentially positive ways, comparing it to how stars are named without direct attribution to astronomers.