AI Has Solved One of Math’s $1 Million Millennium Prize Problems
Mathematicians at OpenAI showed that the Navier-Stokes equations, which describe how fluids flow, can sometimes “blow up.” But the massive result is not without controversy. The post AI Has Solved One of Math’s $1 Million Millennium Prize Problems first appeared on Quanta Magazine.
On September 8, 2026, OpenAI announced that a team of 10,000 autonomous AI agents had solved one of the six remaining Millennium Prize Problems, a set of unsolved math challenges first posed by the Clay Mathematics Institute in 2000. Each problem carries a $1 million prize. The solved problem involves the Navier-Stokes equations, which describe how fluids like water or air move. The AI’s solution was formally verified using the programming language Lean, a tool used to check mathematical proofs with absolute certainty. This marks the first time an AI model has produced a proof of this significance in mathematics. The result, if confirmed, could represent a major shift in how mathematicians approach complex problems using artificial intelligence.
The Navier-Stokes equations are a set of differential equations that describe the motion of fluids, such as ocean currents or air flows. These equations were first formulated in the mid-1800s and are fundamental to fluid mechanics. A key question about these equations is whether their solutions always remain well-behaved or if they can develop singularities—points where fluid velocity becomes infinite, creating chaotic or unpredictable behavior. The Millennium Prize Problem specifically asks whether such singularities can form in three-dimensional space without boundaries. Fluids in the real world are made of molecules, so these mathematical singularities don’t directly apply to physical systems, but they reveal surprising and counterintuitive behaviors in idealized fluid models.
Two independent teams, one led by Tristan Buckmaster at New York University and Levent Alpöge at Anthropic, and another by OpenAI, announced solutions to related problems involving the Navier-Stokes and Euler equations. The Euler equations are a simpler version of the Navier-Stokes equations that describe fluids with no viscosity, or friction. Both teams relied on work by Diego Córdoba of the Institute for Mathematical Sciences in Madrid and Luis Martínez-Zoroa of CUNEF University, who developed new analytical techniques to study fluid behavior. OpenAI used 10,000 AI agents running on an advanced internal model for 88 hours to solve the Navier-Stokes problem, while Buckmaster and Alpöge used AI models to solve a related Euler equation problem. The computational cost for OpenAI’s solution was estimated at several million dollars.
Luis Martínez-Zoroa and Diego Córdoba pioneered a new approach to studying fluid equations that avoids relying on computer simulations. Instead, they created an infinite sequence of layers, each representing a non-singular solution to the equations. By combining these layers in a process they call an infinite cascade, they constructed a new solution that contains a singularity. However, their initial solution did not meet the strict criteria of the Millennium Prize Problem because the combined forcing function, which models external forces like gravity or propellers, was not mathematically smooth. The AI teams later refined this technique to produce a smooth forcing function, satisfying the problem’s requirements.
To ensure the correctness of their mathematical proofs, both teams used Lean, a programming language designed for formal verification. Lean allows mathematicians to write proofs in a way that a computer can check for errors with absolute certainty. This process eliminates human error in manual proof-checking. The AI-generated proofs were first written in natural language and then translated into Lean code, which was verified to confirm that the mathematical statements were logically sound. While Lean ensures the proof is correct, mathematicians must still confirm that the statement being proven matches the original problem, a step that requires human judgment.
The race to solve the Millennium Prize Problems using AI has involved both collaboration and competition. Buckmaster and Alpöge announced their results for the Euler equations just before OpenAI’s announcement. OpenAI acknowledged their priority for the Euler result but claimed the Navier-Stokes solution as their own. Buckmaster expressed frustration that OpenAI may have benefited from their work, while OpenAI stated their solution was inspired by rumors of Buckmaster and Alpöge’s progress. The timeline and contributions of each team remain under discussion, but the intellectual foundation for both solutions traces back to the work of Córdoba and Martínez-Zoroa.
