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. 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. Sébastien Bubeck 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 Tristan Buckmaster at New York University and Levent Alpöge 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. Major breakthrough made on famous Millennium maths problem Buckmaster and Alpöge said they received a “great deal of help from” large language models (LLMs), including LLMs from Anthropic and OpenAI.
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