Anthropic AI model advances Riemann hypothesis proof with $31m compute spend
The breakthrough, initiated by a staff member without mathematical training, reignites debate over authorship and responsibility in the age of autonomous AI research.
Anthropic has announced that an unreleased artificial intelligence model has made significant progress on the Riemann hypothesis, one of the most enduring unsolved problems in mathematics concerning the distribution of prime numbers. The system successfully increased the lower bound of solutions for which the hypothesis holds true, marking a notable advancement in a field that has resisted resolution for more than 150 years.
The initiative began when an Anthropic staff member, who lacked significant mathematical training, prompted the model to “take a real stab” at proving the hypothesis. The model then operated autonomously for 36 hours, coordinating a complex network of 60 sub-agents to test 650 different ideas. The computational effort required 31 million in total resources to execute the extensive search and verification process.
The architecture of the AI’s effort was highly structured. According to a footnote in the resulting paper, two sub-agents were responsible for developing the key mathematical ideas, while 13 contributed concepts to these agents. Thirty sub-agents attempted to develop new ideas but were unable to succeed, 13 served as validators to check argument correctness, and two assisted in writing the initial paper.
Two of Anthropic’s in-house mathematicians confirmed the findings, which were subsequently formalised using the open-source proof assistant Lean. This verification process ensures that the mathematical logic meets rigorous academic standards, even though the initial discovery was driven by an untrained human prompt and autonomous machine execution.
This development follows a series of recent AI-led mathematical breakthroughs, including OpenAI’s internal “Astra” model proving ten major results and a separate Anthropic effort that disproved the Jacobian conjecture. The growing capability of large language models to solve or advance complex mathematical problems has sparked intense discussion within the scientific community.
In June, a group of prominent mathematicians signed a public declaration expressing concern that AI could undermine the value of attributing proofs to specific authors. They argued that true mathematical proofs should be attributable to individuals who assume responsibility for their correctness. However, the field remains divided on how to integrate these new techniques. Fields Medal winner Timothy Gowers has suggested that theorems not being associated with specific mathematicians may not be problematic, comparing the potential shift to how stars are not named after astronomers.
