For more than 150 years, the Riemann hypothesis has stood as one of the foremost unsolved problems in mathematics, a long-running mystery about the distribution of prime numbers. There is presently a $1 million bounty for a working general evidence of the hypothesis, which stays unclaimed.
Contemporary AI models still can’t solve it either — however they can make lots more development than you might anticipate, a finding that’s probably to reopen longstanding questions about contemporary AI’s ability to discover new scientific and mathematical ideas.
On Monday, Anthropic declared that an as-yet-unreleased model had made considerable progress at the Riemann hypothesis, considerably expanding the lower bound of solutions for which the hypothesis holds true.
Even more amazing is how the progress was made: An Anthropic staff member without major mathematical training prompted the model to “take a real stab” at proving the hypothesis, then left the model to coordinate the task throughout the following day and a half.
All informed, the model tested 650 different ideas for solving the problem, coordinating across 60 subagents and spending 31 million output tokens in total.
“Out of the 60 subagents, two have been liable for evolving the main mathematical ideas,” a footnote to the paper describes, “13 contributed ideas to these agents, 30 attempted (however were unable) to develop new ideas, 13 served as validators to check the correctness of the arguments, and the final two assisted 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 duration of this year, and the launch of more powerful models has led to more extraordinary outcomes. OpenAI lately launched a set of 10 big results proved by its internal “Astra” model, while a separate effort from Anthropic disproved the longstanding Jacobian conjecture.
The growing body of outcomes has led to both excitement and concern in the mathematical field. In a public announcement signed in June, a set of prominent mathematicians triggered concerns that AI could undermine vital values of the field — specifically the standard that true mathematical proofs ought to be “attributable to precise authors who take credit for their discovery and expect responsibility for their correctness.”
But the field continues to split on how mathematicians how approach the new research strategies. In a blog post responding to the announcement, 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 wherein mathematical theorems are no longer related to 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.
