Anthropic announced Monday that an as-yet-unreleased AI model made significant progress on the Riemann hypothesis, one of mathematics’ oldest and most famous unsolved problems. The model, which has not been publicly named, increased the lower bound of solutions for which the hypothesis holds true — a meaningful step forward on a problem that has resisted a general proof for more than 150 years.
The result is already drawing attention not just for the mathematical advance, but for how it was achieved. An Anthropic staff member with no significant mathematical training prompted the model to “take a real stab” at proving the hypothesis, 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 roughly $31 million in compute.
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According to a footnote in the paper describing the work, “Out of the 60 subagents, two were responsible for developing the key mathematical ideas, 13 contributed ideas to these agents, 30 attempted (but were unable) to develop new ideas, 13 served as validators to check the correctness of the arguments, and the final two helped to write the initial paper.” The finding was confirmed by two of Anthropic’s in-house mathematicians and formalized using the open-source proof assistant Lean.
Why this matters for the field of mathematics
The Riemann hypothesis, first proposed by Bernhard Riemann in 1859, concerns the distribution of prime numbers and has deep implications across number theory and cryptography. A working general proof carries a $1 million bounty from the Clay Mathematics Institute, which remains unclaimed. While contemporary AI models still cannot solve the hypothesis outright, the progress demonstrated by Anthropic’s model suggests they are capable of generating novel mathematical ideas — a capability that was largely theoretical until recently.
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This is part of a broader string of mathematical breakthroughs attributed to large language models. A number of Erdős problems have been solved by AI models over the course of this year, and the release of more powerful models has led to more impressive results. OpenAI recently released a set of ten major results proved by its internal “Astra” model, while a separate effort from Anthropic disproved the long-standing Jacobian conjecture.
The growing body of results has sparked both excitement and concern in the mathematical community. 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 true mathematical proofs should be “attributable to specific authors who take credit for their discovery and assume responsibility for their correctness.”
Mixed reactions from mathematicians
The field remains split on how to approach these new research techniques. In a blog post responding to the declaration, Fields Medal winner Timothy Gowers questioned whether the influence of AI might 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.
For now, the Anthropic result stands as a demonstration of what autonomous AI systems can achieve when given the freedom to explore a problem without human intervention. The use of Lean for formal verification also addresses one of the key concerns about AI-generated proofs — their reliability — by providing a machine-checkable guarantee of correctness.
The announcement is likely to intensify the debate over how mathematics should be practiced in an era of increasingly capable AI. As models continue to improve, the question may shift from whether AI can contribute to mathematics to how the field should adapt to a new kind of collaborator.
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