OpenAI says it has solved a million-dollar math problem

OpenAI claims that one of its artificial intelligence models has found the solution to one of the most difficult mathematical problems in the world, which remained unsolved for decades and accompanied by a million dollar prize. But before we can talk about the problem being definitively solved, despite the announcements and articles, the severe and final judgment of the scientific community will be needed.

The company led by Sam Altman has announced that it has obtained a solution to the so-called problem of the existence and regularity of the Navier-Stokes equations, one of the seven “Millennium Prize Problems” identified in 2000 by the Clay Mathematics Institute. OpenAI published a description of the result along with a demonstration formalized in the Lean language.

We’re sharing a solution to the Navier-Stokes Millennium Prize Problem, one of the deepest problems at the frontier of mathematics.

The proof was produced by a group of agents, using an OpenAI next-generation model significantly more capable than GPT-6 Astra.

The problem… pic.twitter.com/8zol3BPTL4

— OpenAI (@OpenAI) September 8, 2026

What are Navier-Stokes equations

Behind an extremely complex mathematical problem there is something very concrete: the movement of fluids. The Navier-Stokes equations are used to describe the behavior of liquids and gases and are fundamental in fields ranging from meteorology to aerodynamics. The problem is not to ask whether the equations work in the many cases in which they are already used, but to understand what might happen mathematically to their solutions in three dimensions. In particular, for decades mathematicians have been trying to establish whether, starting from certain initial conditions, the solutions always remain regular or can develop a singularity, a “blow-up”, in a finite time.

OpenAI claims to have demonstrated precisely this second possibility: conditions would exist in which the equations lead to a singularity. “Our evidence shows that there are fluids that start in perfectly normal conditions and that, according to the Navier-Stokes equations, actually reach an infinite velocity in a finite time,” Ven Chandrasekaran, a researcher at OpenAI, explained to Nature. It does not mean that a real fluid can reach infinite speed. The result would concern the behavior of the mathematical model and would show that, under certain conditions, the equations can reach a point where they no longer describe a regular solution.

Ten thousand AI agents working on the problem

The way OpenAI claims to have arrived at the result is also significant. According to the reconstruction of Nature and Quanta Magazine, the company used thousands of artificial intelligence agents that worked in parallel on the problem. After an initial phase conducted on a simplified version with around a thousand agents, OpenAI would have drastically increased the resources dedicated to research. To address the complete Navier-Stokes problem, approximately 10,000 agents would be employed based on an internal model not yet available to the public.

According to Quanta Magazine, after 88 hours the system would have produced the demonstration of the existence of the singularity; another 17 hours would have been needed to formalize the result. Overall, the agents exchanged almost five million messages. Sébastien Bubeck, a mathematician at OpenAI, estimated the computational cost of the operation at several million dollars.

The million dollars is not yet won

The result concerns one of the seven Millennium Prize Problems, the great mathematical challenges selected by the Clay Mathematics Institute at the beginning of the millennium. For each solution there is a prize of one million dollars. However, this does not mean that OpenAI has already won the prize. The rules of the Clay Mathematics Institute are particularly strict. A solution must be published in a qualified scientific venueat least two years must pass from publication and the result must gain general acceptance by the international mathematical community before Clay can consider it for the award of the prize.

Martin Bridson, president of the Clay Mathematics Institute, welcomed the announcement with interest but without certifying the result. “It is certainly an exciting day as we reflect on the announcement of major advances in human understanding of mathematics,” he told Nature. Caution is therefore essential. OpenAI has presented a solution and a formalized demonstration, but independent scientific verification has only just begun.

A crucial technical distinction: forced or unforced

There is also a technical aspect that reduces the scope of the announcement. OpenAI’s result concerns the “forced” version of the Navier-Stokes equationsthat is, a fluid subjected to an external force applied in a controlled manner. The official criteria set by the Clay Mathematics Institute for awarding the prize concern the “unforced” version of the problem, the one without external forces applied. According to several specialized publications, this distinction means that the result announced by OpenAI, however relevant, would not in itself satisfy the criteria required for the Millennium Prize.

The race between mathematicians and artificial intelligence

The story also contains a scientific competition yet to be clarified, and in the days following the announcement it turned into a real public dispute. OpenAI decided to focus its resources on the problem after learning that i mathematician Levent Alpögefrom Harvard University, e Tristan Buckmasterfrom New York University, were obtaining important results in the same field also using artificial intelligence systems. Alpöge, in addition to being a Junior Fellow at Harvard, is also an employee of Anthropic, OpenAI’s rival company.

Alpöge and Buckmaster released on September 7th a work related to Euler’s equationswhich can be considered a viscosity-free version of the problem. They also used Claude from Anthropic and models from OpenAI in their work. The two researchers also said they had obtained results on the more general problem, intended to be published later.

Buckmaster later published a public statement accusing OpenAI of having learned of his and Alpöge’s work through rumors in the days preceding the announcement. According to Buckmaster, an OpenAI researcher, Sébastien Bubeck, asked him twice to exclude Alpöge from the authors of the paper precisely because he was an employee of Anthropic, going so far as to ask him, when faced with his refusal, “why would you want to ruin your career?”. OpenAI rejected the accusations: in a press conference, Bubeck denied the rumors about the origin of the proof, claiming that OpenAI’s internal system had solved the Euler problem with a completely independent method, while admitting that the solution to the complete Navier-Stokes problem followed a similar path to that of Buckmaster and Alpöge.

Some aspects of the chronology and interactions between the researchers and OpenAI therefore remain to be clarified, including the question of how much previous work contributed to the path that led to the result announced by the company. The story therefore opens a second front in addition to the strictly mathematical one: establishing what role human researchers, previous work and artificial intelligence systems have had respectively, and who should receive credit for them. If OpenAI’s demonstration passes the scrutiny of the scientific community, the result could still represent an important step in the history of artificial intelligence applied to research. No longer just systems capable of summarizing existing literature, assisting researchers or solving complex exercises, but tools used to attempt to directly move the frontier of mathematical knowledge.