The Riemann Hypothesis has been one of the major unsolved problems in mathematics for more than 150 years, and is a long-standing puzzle about the distribution of prime numbers. There is currently a $1 million bounty on the table for a practical general proof of the hypothesis, but it has not yet been claimed.
Modern AI models still cannot solve this problem. But it could be much more advanced than expected, and the discovery could once again clarify long-standing questions about modern AI’s ability to discover new scientific and mathematical ideas.
On Monday, Anthropic announced that its yet-to-be-released model has made significant progress on the Riemann conjecture, significantly increasing the lower bound of solutions for which the hypothesis holds.
Even more impressive is the progress. An Anthropic staff member with limited mathematical training encouraged the model to “get serious” about proving the hypothesis, then left it alone to tweak the task for a day and a half.
Overall, the model tested 650 different ideas to solve the problem, coordinated across 60 subagents, and spent a total of 31 million output tokens.
A footnote to the paper explains that “out of the 60 subagents, two were responsible for developing important mathematical ideas.” “Thirteen people contributed ideas to these agents, 30 tried (but failed) to develop new ideas, 13 acted as validators checking the correctness of the arguments, and the last two helped write the first paper.”
This discovery 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 series of mathematical advances led by large-scale language models (LLMs). Throughout this year, many problems in Erdos have been solved by AI models, and the release of more powerful models has led to more impressive results. OpenAI recently released a key set of 10 results proven by its internal “Astra” model. Meanwhile, another effort by Anthropic has disproved the long-standing Jacobian conjecture.
The growing number of results is causing both excitement and concern in the mathematics field. In a public declaration signed in June, a group of prominent mathematicians expressed concern that AI could undermine important values in the field, particularly the standard that true mathematical proofs “should be attributed to a specific author who takes credit for the discovery and is responsible for its correctness.”
However, the field remains divided over how mathematicians should approach new research methods. In a blog post responding to the declaration, Fields Medal winner Timothy Gowers questioned whether mathematics would become more complex and positive under the influence of AI.
“If mathematical theorems reached a world independent of mathematicians, perhaps that would be less of a problem than the fact that stars are not named by astronomers, and most stars are not named at all,” Gowers wrote.
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