OpenAI says AI model solved one of mathematics' Millennium Prize Problems

OpenAI said on Tuesday that an unreleased artificial intelligence model has solved the Navier-Stokes existence and smoothness problem, one of mathematics' seven famed Millennium Prize Problems.

The company said the model took 88 hours to produce the solution, deploying as many as 10,000 AI agents working in parallel, in what could mark one of the most significant demonstrations yet of AI's growing ability to tackle advanced mathematical research.

The Navier-Stokes problem concerns equations used to describe the motion of fluids such as water and air, and asks, broadly, whether their solutions always remain well-behaved or can develop singularities under certain conditions.

OpenAI's proof claims to demonstrate such a breakdown, or "blowup," providing an answer to a problem that has remained unresolved for decades.

The problem is one of seven Millennium Prize Problems designated by the Clay Mathematics Institute in 2000, with a $1 million prize offered for a correct solution to each. Only one of the seven had previously been solved.

OpenAI also released a paper detailing its proposed solution, including a proof written in Lean, a programming language and theorem-proving system used to formally verify mathematical arguments.

The claim, however, will require scrutiny by the wider mathematical community before the problem can be considered definitively settled.

The announcement comes amid a dispute about parallel research by New York University mathematician Tristan Buckmaster and Levent Alpoge, a mathematician working at rival AI company Anthropic. OpenAI acknowledged that it intensified its work on the problem after learning that other mathematicians were pursuing a similar direction, but said it had not seen their work.

The development highlights the rapidly expanding role of AI in higher mathematics, after AI systems in the last year solved several longstanding problems that had resisted mathematicians for years or decades.