The Riemann hypothesis has remained unsolved for over 150 years, yet an unreleased Anthropic model has pushed the lower bound of its known solutions further than before.
A working general proof of the hypothesis carries a $1 million prize that remains unclaimed. Current AI systems cannot solve it, but they can now advance the search in ways that challenge assumptions about machine discovery.
How the work was done
On Monday, Anthropic announced the result. An internal staff member with no significant mathematical training prompted the model to attempt a proof. The human then stepped back while the system coordinated the work for 36 hours.
The model evaluated 650 different approaches. It managed 60 sub-agents and consumed 31 million tokens in total.
According to a footnote in the paper, two sub-agents developed the core mathematical ideas. Thirteen others contributed to these agents. Thirty tried to develop new ideas but failed. Thirteen acted as validators to check the arguments. The final two helped write the initial paper.
Two in-house mathematicians confirmed the finding. The work was formalised using the open-source proof assistant Lean.
A pattern of results
This follows a series of mathematical breakthroughs led by large language models. Several Erdos problems were solved by AI this year. More powerful models have produced more impressive results.
OpenAI recently released a set of 10 major results proved by its internal Astra model. A separate effort from Anthropic disproved the long-standing Jacobian conjecture.
Reactions from the field
The growing body of results has caused both excitement and concern. In June, a group of prominent mathematicians signed a declaration. They argued that AI could undermine critical values of the field. Specifically, they noted that true mathematical proofs should be attributable to specific authors who take credit for their discovery and assume responsibility for their correctness.
The community remains divided on how to approach these new techniques. Fields Medal winner Timothy Gowers responded to the declaration in a blog post. He questioned whether the influence of AI might change mathematics in a more complex and positive way.
“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,” Gowers wrote.
