Anthropic's Unreleased AI Tackles Major Unsolved Math Problem

For over a century and a half, the Riemann hypothesis has remained one of mathematics' most profound unsolved puzzles, a persistent enigma concerning the distribution of prime numbers. A substantial $1 million bounty currently awaits a verifiable general proof of this hypothesis, a reward that has yet to be claimed.

While contemporary AI models have not yet fully cracked this challenge, they are demonstrating significantly more progress than might be anticipated. This development is poised to reignite long-standing discussions regarding the capacity of modern AI to pioneer new scientific and mathematical concepts.

This past Monday, Anthropic disclosed that an unreleased model had achieved notable progress on the Riemann hypothesis, successfully elevating the lower bound of solutions for which the hypothesis is confirmed to be true.

Even more remarkable is the methodology behind this advancement: an Anthropic team member, who lacked extensive mathematical expertise, simply instructed the model to "take a real stab" at proving the hypothesis. The model was then left to independently manage and execute the task over the subsequent day and a half.

In total, the model explored 650 distinct approaches to the problem, coordinating its efforts across 60 specialized sub-agents and expending a total of 31 million computational units in the process.

As detailed in a footnote to the accompanying paper, "Out of the 60 subagents, two were responsible for developing the key mathematical ideas, 13 contributed ideas to these agents, 30 attempted (but were unable) to develop new ideas, 13 served as validators to check the correctness of the arguments, and the final two helped to write the initial paper."

The breakthrough was subsequently validated by two of Anthropic's in-house mathematicians and formally structured using Lean, an open-source proof assistant.

This achievement marks the latest in a series of mathematical breakthroughs driven by Large Language Models (LLMs). This year alone has seen AI models resolve several Erdos problems, with the deployment of increasingly sophisticated models yielding even more impressive outcomes. OpenAI recently unveiled ten significant mathematical results proven by its internal "Astra" model, while a separate initiative by Anthropic successfully disproved the long-standing Jacobian conjecture.

This accumulating body of results has sparked both enthusiasm and apprehension within the mathematical community. In a public declaration issued in June, a collective of prominent mathematicians voiced concerns that AI could erode fundamental values of the discipline, particularly the expectation 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 novel research methodologies. In a blog post responding to the declaration, Fields Medal laureate Timothy Gowers contemplated whether AI's influence might, in fact, transform mathematics in a more intricate and beneficial manner.

Gowers mused, "If we arrive at a world where mathematical theorems are no longer associated with 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."