Claude and Astra on open mathematics

OpenAI confirmed it solved the Navier-Stokes Millennium Prize Problem using a multi-agent system of roughly 10,000 concurrent agents working for about 88 hours starting September 1, with Lean formalization and verification taking an additional 17 hours via GPT-6 Astra. Across all attempted Millennium Prize problems, the agents sent 4.9 million messages and used about 300 billion output tokens. Researcher Sébastien Bubeck estimated the computational cost at several million dollars. Only one of the seven Millennium Prize Problems had ever previously been solved before OpenAI's announcement.

OpenAI also stated it made substantial progress on a second unnamed Millennium Prize problem and was working out how to share the result. An unnamed source cited by The Information expects the Hodge Conjecture to be solved soon; claims that an internal model called Aeon is responsible remain unconfirmed by OpenAI. A new OpenAI model reportedly began training on August 28 with mathematical capabilities said to significantly exceed GPT-6 Astra.

Separately, OpenAI's internal Astra model resolved or substantially advanced ten open problems in mathematics and computer science, including constructing a non-sofic group, disproving Connes's rigidity conjecture, and settling several Erdős problems. OpenAI researcher Noam Brown acknowledged unsuccessful attempts including on Millennium Prize problems, and said the team had not spent heavily on each problem. Anthropic researcher Levent Alpöge reproduced half of Astra's ten results within 24 hours using a different model called Fable.

Anthropic researchers Levent Alpöge and Ava Howell used an internal, non-public Claude variant to find elliptic curves with ranks of at least 30 and 31, surpassing the rank-29 record set in August 2024. The prior progression from rank 28 to rank 29 had taken more than 18 years; the new results emerged within days from a relatively simple prompt. Both researchers have extensive background in elliptic curve research.