For over a century and a half, the Riemann hypothesis has remained one of mathematics’ most elusive unsolved problems, concerning the distribution of prime numbers. A $1 million reward awaits anyone who can provide a general proof, yet it remains unclaimed.
Recently, an unreleased AI model developed by Anthropic has made notable strides on this hypothesis. The model significantly increased the lower bound of solutions where the hypothesis holds true. Remarkably, this progress was achieved when an Anthropic team member, without extensive mathematical training, instructed the model to attempt a proof. Over the next 36 hours, the AI coordinated 60 sub-agents, testing 650 different approaches, and expending 31 million tokens in the process.
Among these sub-agents, two were pivotal in developing key mathematical concepts, 13 contributed ideas, 30 attempted new ideas without success, 13 served as validators, and two assisted in drafting the initial paper. The findings were validated by Anthropic’s in-house mathematicians and formalized using the Lean proof assistant.
This development is part of a broader trend where large language models (LLMs) are making significant contributions to mathematics. Earlier this year, AI models solved several Erdős problems, and OpenAI’s “Astra” model recently proved ten major results. Additionally, another Anthropic model disproved the longstanding Jacobian conjecture.
These advancements have sparked both excitement and concern within the mathematical community. In June, a group of prominent mathematicians expressed worries that AI could undermine the field’s values, particularly the attribution of proofs to specific individuals. However, others, like Fields Medalist Timothy Gowers, suggest that AI’s role might transform mathematics in complex and potentially positive ways.
Anthropic’s recent achievement underscores the evolving capabilities of AI in tackling complex mathematical problems. As AI continues to advance, it challenges traditional notions of authorship and proof in mathematics, prompting the community to reconsider its methodologies and standards.
