Anthropic's unreleased model expands lower bound of Riemann hypothesis

The Riemann hypothesis, a central open problem in mathematics for more than 150 years that concerns the distribution of prime numbers, carries a one-million-dollar prize that has not yet been claimed. Anthropic announced that an unreleased model succeeded in advancing the hypothesis significantly by raising the lower bound of the solutions for which the hypothesis holds, a result that amounts to a proof of a broader special case rather than a full general proof that would earn the monetary award.

According to the report, a non-specialist employee at Anthropic asked the model to attempt a proof of the hypothesis, after which the system operated autonomously for a day and a half. During that period the model examined 650 distinct ideas, coordinated its work with 60 subagents, and consumed 31 million tokens of output in total. A footnote in the paper specifies that two subagents were responsible for developing the core mathematical ideas, 13 contributed ideas, 30 attempted without success to develop new directions, 13 served as validators that checked the correctness of the arguments, and the remaining two helped write the paper itself.

The outcome was validated by two Anthropic mathematicians and formalized in Lean, an open-source proof-verification language, thereby providing a logical guarantee that the proof is sound and not reliant on statistical inference from the language model. The paper notes, however, that the work appears only as a preprint that has not undergone peer review, so the findings remain valid at the laboratory level only.

This advance joins a series of recent mathematical breakthroughs achieved by LLMs. Several Erdos problems have been solved by AI models, OpenAI recently released ten central results proven by its internal Astra model, and a separate Anthropic effort refuted the long-standing Jacobian conjecture. The trend revives questions about the capacity of models to uncover new scientific and mathematical ideas.

A growing body of results also raises concern within the mathematical community. In a public statement signed in June, a group of senior mathematicians expressed worry that AI could erode core values of the discipline, particularly the norm that genuine proofs should be attributed to specific authors who assume credit and responsibility for their correctness. The community remains divided over how to engage with these new research techniques.

In response, Fields Medalist Timothy Gowers questioned the assumption that AI’s impact is necessarily negative, stating, “if we reach a world where mathematical theorems are no longer associated with mathematicians, perhaps that would not be more problematic than the fact that stars are not named after astronomers and most are not named after anyone.”