Anthropic's AI model makes significant progress on the 150-year-old Riemann Hypothesis

The Riemann Hypothesis, one of the greatest problems in mathematics that has remained unsolved for over 150 years and carries a 1 million dollar prize, has entered the radar of artificial intelligence. An AI model developed by Anthropic, which has not yet been released to the public, managed to significantly raise the known lower bound for solutions where the hypothesis holds true, even though it did not solve the problem entirely.

650 DIFFERENT IDEAS AND 60 SUB-AGENTS WERE USED

The process began when an Anthropic employee without advanced mathematical training asked the model to prove the Riemann Hypothesis. Working non-stop for about a day and a half, the system coordinated the work by establishing a massive collaborative network.

Throughout the study, 31 million output tokens were generated, 650 different solution ideas were tested, and 60 different sub-AI agents were involved in the process. Two sub-agents developed fundamental mathematical ideas, while 13 agents provided supporting ideas. Although 30 agents attempted to generate new arguments, they were unsuccessful. Of the remaining agents, 13 verified the accuracy of the generated arguments, and the final two ensured the work was compiled into a paper.

RESULTS VERIFIED BY MATHEMATICIANS

The findings reached by the model were reviewed and approved by two human mathematicians working at Anthropic. The results obtained were also formatted into an official structure using Lean, an open-source mathematical proof assistant.

The 1 million dollar prize, which will be awarded to anyone who can provide a general proof of the Riemann Hypothesis—which is directly linked to the distribution of prime numbers—remains unclaimed.

THE ROLE OF AI IN MATHEMATICS SPARKS DEBATE

This move by Anthropic has reignited debates regarding the role of large language models in scientific and mathematical research. Earlier this year, different AI systems were used to solve Erdős problems, and OpenAI's internal Astra model successfully proved 10 significant mathematical results. In another study by Anthropic, it was demonstrated that the long-debated Jacobian conjecture was false.

This rapid rise of AI in mathematical research has divided experts in the field. In a statement published by prominent mathematicians in June, a warning was issued that AI could harm traditional proof processes. The statement emphasized the importance of attributing mathematical discoveries to the actual authors who take responsibility for them and receive credit for the success.