Secret Anthropic AI Cracks Century-Old Math Enigma

For over a century and a half, the Riemann hypothesis has remained one of mathematics' most formidable unsolved challenges, representing a profound enigma concerning the distribution of prime numbers. Despite a standing $1 million reward for a comprehensive proof, the prize remains unclaimed. Intriguingly, while contemporary AI models haven't fully solved it, recent advancements suggest they are making significant strides, a development that is poised to reignite discussions about AI's capacity for novel scientific and mathematical discoveries.

Anthropic recently announced a notable breakthrough achieved by an unreleased AI model concerning the Riemann hypothesis. This model substantially increased the lower bound of solutions for which the hypothesis holds true. What makes this achievement particularly impressive is the methodology: an Anthropic staff member, lacking extensive mathematical expertise, simply instructed the model to 'take a real stab' at proving the hypothesis. The model then independently orchestrated the task over a day and a half, exploring 650 distinct problem-solving approaches. It coordinated the efforts of 60 subagents and processed a staggering 31 million output tokens.

A detailed footnote in the research paper clarified the roles of these subagents: two were instrumental in conceiving the core mathematical ideas, 13 provided supportive ideas to these key agents, 30 attempted but failed to generate new concepts, 13 acted as validators to verify the accuracy of arguments, and the final two assisted in drafting the initial paper. This intricate process was subsequently verified by two of Anthropic's internal mathematicians and formally documented using the open-source proof assistant, Lean.

This accomplishment adds to a growing list of mathematical breakthroughs powered by large language models (LLMs). Throughout the current year, AI models have successfully resolved several Erdos problems, with the continuous release of more powerful models leading to increasingly impressive outcomes. OpenAI's internal 'Astra' model, for instance, recently presented 10 major mathematical proofs, while a separate initiative by Anthropic managed to disprove the long-standing Jacobian conjecture.

The burgeoning catalog of AI-driven mathematical results has elicited a mix of excitement and apprehension within the mathematical community. In a public statement issued in June, a collective of prominent mathematicians voiced concerns that AI could erode fundamental values of the field, specifically the principle that genuine mathematical proofs should be 'attributable to specific authors who take credit for their discovery and assume responsibility for their correctness.' However, the field remains divided on the optimal approach to these emerging research techniques. In a blog post responding to the declaration, Fields Medalist Timothy Gowers pondered whether AI's influence might, in fact, transform mathematics in a more nuanced and beneficial manner, suggesting that a world where theorems are not solely attributed to individual mathematicians might be no more problematic than stars not being named after astronomers.