For more than 150 years, the Riemann hypothesis has stood as one of the major unsolved problems in mathematics, a long-running mystery about the distribution of prime numbers. There is currently a $1 million bounty for a working general proof of the hypothesis, which remains unclaimed.
Contemporary AI models still can’t solve it either — but they can make a lot more progress than you might expect, a finding that’s likely to reopen long-standing questions about contemporary AI’s ability to discover new scientific and mathematical ideas.
On Monday, Anthropic announced that an as-yet-unreleased model had made significant progress on the Riemann hypothesis, significantly increasing the lower bound of solutions for which the hypothesis holds true.
Even more impressive is how the progress was made: An Anthropic staff member without significant mathematical training prompted the model to “take a real stab” at proving the hypothesis, then left the model to coordinate the task across the following day and a half.
All told, the model tested 650 different ideas for solving the problem, coordinating across 60 sub-agents and spending 31 million in total.
“Out of the 60 subagents, two were responsible for developing the key mathematical ideas,” a footnote to the paper explains, “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.
This is the latest in a string of mathematical breakthroughs led by Large Language Models, or LLMs. A number of Erdos 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 caused both excitement and concern in the mathematical field. 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.”
But the field is still split on how mathematicians should approach the 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.
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