OpenAI claims one of its AI models has found a solution to the Navier-Stokes problem, one of the toughest and most enduring puzzles in mathematics, sitting on the list of Millennium Prize Problems, solutions for which come with a $1 million reward. The announcement came after an unusual flurry of activity around the problem, along with a dispute around how the work came about.
The Navier-Stokes equations describe how fluids move in space, such as air over an aircraft wing or water out of a tap. Though the equations were first written down around 200 years ago, they aren’t well understood. The question posed by the Clay Mathematics Institute for the Millennium Prize Problem is to determine if the equations always work or if there are situations where they stop making sense and start spouting nonsense, what mathematicians call “blow-ups”.
To settle the question, OpenAI first set 1000 AI agents on the Euler problem, which is a cousin of the Navier-Stokes problem and a step on the path to a full solution. It took the agents 50 hours to find blow-ups in this case. They then set 10,000 agents the task of extending the blow-ups to apply to the full Navier-Stokes problem, which took just 11 hours to finish.
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OpenAI said during a press conference that if a customer wanted to run the same problem, it would cost around $15 million. OpenAI didn’t name the AI model used, but said it was “significantly more capable” than even its latest GPT-6 Astra model.
“This problem has remained unsolved for 200 years because the Navier-Stokes equations are just so enormously complex, and the pen-and-paper calculations you need to do in order to solve this problem are just mind-bogglingly intricate,” says Venkat Chandrasekaran at OpenAI, who was also on the press conference call.
at OpenAI says the news is the “spectacular combination of the arc we have seen over the last 12 months”.
The result is the latest in a string of shocking mathematical discoveries led by AI in recent months. In May, an OpenAI model cracked a decades-old conjecture by Paul Erdős, causing a stir in mathematical circles. Later, the Claude Fable 5 AI found a counterexample to the Jacobian conjecture, which had stood for nearly a century. Last week, an AI model formalised Fermat’s last theorem in just 11 days.
OpenAI’s latest groundbreaking mathematical discovery came just hours after  at New York University and  at AI company Anthropic announced that they had cracked three significant problems considered “stepping stones” to the Navier-Stokes problem.
Buckmaster and Alpöge also showed that blow-ups can appear in Euler equations; OpenAI has now shown that the same is true of Navier-Stokes.
Buckmaster and Alpöge said they received a “great deal of help from” large language models (LLMs), including LLMs from Anthropic and OpenAI. But when rumours suggested that OpenAI had gone one step further and solved the wider Navier-Stokes puzzle – something the company formally announced just hours later – Buckmaster released a statement suggesting that OpenAI acted unusually and opaquely when he approached the company for clarification.
Buckmaster wrote in a that, as a result of these rumours, he emailed a prominent mathematician at OpenAI to clarify matters. He says that in a subsequent meeting with OpenAI staff, he was given no hard details about how OpenAI achieved the result, but he took what little he was told as a “red flag”, as it appeared to describe the same technique and path that he and Levent had followed to achieve their interim findings.Â
According to Buckmaster, OpenAI then admitted it hadn’t started its own AI search for a solution until after it had heard of his and Alpöge’s work, and didn’t respond to questions about whether its AI model had been trained or fine-tuned on that existing work – which was being stored in OpenAI’s Codex model, where the company would theoretically have access to it. The pair had used Codex as a customer, not a research partner, so expected that their work should remain secure and private.
Buckmaster makes it clear in his version of events that he is making no claims or accusations about how OpenAI arrived at its solution, saying only that he wished the focus could instead be on the mathematics rather than scandal and intrigue. “I have not seen OpenAI’s proof. I do not know what their model did, or how. I do not know whether our data was used. I am not accusing anyone of anything,” he wrote in the posted document.
In a press conference announcing its discovery, OpenAI categorically denied using Buckmaster and Alpöge’s proof or prompts in its work. It also said that its model’s proof of Euler is different to that put forward by Buckmaster and Alpöge.Â
It also said that no humans accessed Codex to see work in progress within. Mark Chen at OpenAI, when asked if that also applied to any of the thousands of AI agents that had worked on the problem, said “that’s also our understanding”.
Though the solution to this Navier-Stokes problem has been a long time coming, AI-generated proofs are often hard to understand and rarely bring the same level of new insight than those created by humans. “There’s been this very strange and unprecedented decoupling, this year alone, between getting answers and getting understanding,” says at University of California, Los Angeles. He says that AI is coming up with new results so quickly that there is not enough time for the “slow, deliberate discussion” needed to unpack them.
Something that will happen slowly is determining how the $1 million Millennium Prize money will be awarded. “The process of evaluation is deliberately unhurried, and we shall ensure that it is absolutely rigorous,” wrote , president of the Clay Mathematics Institute, to Â鶹´«Ă˝ in an email.
