An unreleased Anthropic model made progress on one of math’s biggest unsolved problems
For more than 150 years, the Riemann hypothesis has stood as one of the major unsolved problems in mathematics. Anthropic hasn't solved it — but the company's models made more progress than you might expect.
For over 150 years, the Riemann hypothesis has remained one of the most elusive problems in mathematics, a mystery surrounding the distribution of prime numbers. While no one has claimed the $1 million prize for a working proof, recent developments suggest that contemporary AI models may be more capable than expected. On Monday, Anthropic announced that an unreleased model had made significant progress on the Riemann hypothesis, boosting the lower bound of solutions for which the hypothesis holds true.
Remarkably, this progress was achieved by a staff member with minimal mathematical background who simply gave the model instructions to "take a real stab" at solving the problem, then left it to coordinate efforts over the following day and a half. The model generated and tested 650 different ideas, using 60 sub-agents to coordinate and spend a total of 31 million hours.
Two sub-agents were responsible for the key mathematical ideas, with 13 contributing to these agents, 30 attempting but failing to develop new ideas, 13 validating the correctness of arguments, and two assisting in writing the initial paper.
The results were validated by two of Anthropic's in-house mathematicians and formalized using the open-source proof assistant Lean. This breakthrough follows a series of mathematical discoveries made by Large Language Models, or LLMs, in recent months. These include several Erdos problems solved by AI models and the release of more powerful models leading to even more impressive results.
OpenAI recently released ten major results proved by its internal "Astra" model, while Anthropic disproved the long-standing Jacobian conjecture.
The growing body of results has generated both excitement and concern within the mathematical community. In a public declaration, a group of prominent mathematicians expressed worry that AI could undermine the field's critical values, particularly the standard of mathematical proofs being attributable to specific authors. However, the field remains divided on how to approach these new research techniques.
In a response to the declaration, Fields Medal winner Timothy Gowers questioned whether AI might change mathematics in a more complex and positive way, suggesting that if mathematical theorems are no longer associated with mathematicians, it might not be problematic, similar to how stars aren't named after astronomers.
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