OpenAI says AI system solved a Millennium Prize math problem in 88 hours

The result has not yet been independently verified or accepted by the Clay Mathematics Institute, which offers a $1 million prize for a solution to each of the problems.

The Navier–Stokes equations describe the motion of fluids and are used in fields ranging from aircraft design and weather forecasting to the study of blood flow. A central question about the equations - whether smooth three-dimensional fluid motion can develop a singularity in finite time - has remained unresolved for roughly 90 years.

OpenAI said its internal model produced both an analytical proof and a formalized version in the Lean proof assistant showing that an initially smooth fluid can develop such a singularity.

The company said the model is "significantly more capable" than GPT-6 Astra and remains under development.

OpenAI launched the effort on Sept. 1 after hearing rumors that two Millennium Prize problems may have been solved. Around 10,000 AI agents worked on the Navier–Stokes problem, exchanging 2.7 million messages and generating about 130 billion output tokens before reaching a result on Sept. 5. Formalization and verification in Lean took another 17 hours.

Based on OpenAI's published pricing for its most advanced models, the computing effort would have cost roughly $10 million, according to the BBC.

The announcement has also prompted a dispute over how the project began.

New York University mathematics professor Tristan Buckmaster said he and Levent Alpöge, a mathematician at OpenAI rival Anthropic, had been working on a related problem using OpenAI's Codex. Buckmaster said he learned on Sept. 3 that information about their progress had been passed to OpenAI and questioned the timing of the company's work.

OpenAI said it began its effort after hearing a rumor that it later realized was related to Buckmaster and Alpöge, but denied accessing their specific work or user data while solving the problem.

The company said it "cannot rule out" the possibility that de-identified data derived from their use of OpenAI products had helped improve its models, while stressing that the mathematical results produced by the two teams differ.

OpenAI said it does not intend to claim the $1 million Millennium Prize, describing the result instead as evidence of rapid progress in its AI systems.