OpenAI announced on Tuesday that it has resolved a significant portion of the Navier-Stokes existence and smoothness problem, a longstanding mathematical challenge that has resisted complete proof for 90 years. The company deployed approximately 10,000 autonomous AI agents powered by a newly developed internal model to attack the problem, reaching a solution in roughly 88 hours.
The Navier-Stokes equations describe fluid dynamics and remain central to understanding turbulence, a phenomenon that scientists continue to study intensively. The Clay Mathematics Institute offers a $1 million Millennium Prize for complete solutions to certain mathematical problems, though OpenAI stated it does not intend to claim the prize for this work. The company’s solution addressed two of the four required statements in the complete proof.
The announcement has generated controversy regarding the circumstances surrounding the work. Tristan Buckmaster, a mathematician at New York University, stated that he and Anthropic researcher Levent Alpöge had been independently working on the same problem using OpenAI’s Codex tool. Buckmaster asserted that information about their progress reached OpenAI on 3 September, hours before the company published its findings. He questioned the timing of OpenAI’s effort, noting the company had only begun working on the problem after learning of their concurrent research.
OpenAI responded by acknowledging the concurrent work and stating it had not accessed any of Buckmaster and Alpöge’s proprietary research through conventional means. The company noted that while it could not entirely rule out that de-identified usage data from their platform interactions contributed to model improvements, the two proofs differ significantly in their approaches and final results.
The company indicated that it became aware of rumors regarding resolved Millennium Prize problems on 1 September and subsequently directed its newly trained model toward addressing several outstanding mathematical challenges. The computational effort required millions of exchanged messages and approximately 130 billion output tokens, representing an estimated cost of roughly $10 million at standard pricing rates.
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