AI solves one of math’s hardest problems in 88 hours, says OpenAI
September 8, 2026 9:52 pm
OpenAI says an internal artificial intelligence system has produced a solution to the Navier-Stokes existence and smoothness problem in about 88 hours, after deploying a system involving roughly 10,000 concurrent AI agents.
In a research release published on Tuesday, OpenAI said it was sharing “a solution to the Navier–Stokes existence and smoothness problem, one of the Millennium Prize Problems.”
PUNCH Online reports that the Navier-Stokes problem is one of seven Millennium Prize Problems established by the Clay Mathematics Institute in 2000 to highlight some of the most difficult mathematical questions that had resisted solution for years.
CMI set aside a $7 million prize fund, with $1 million allocated to the solution of each problem.
The OpenAI development comes after the company said its internal AI system was deployed on the problem as part of a wider effort to test its capabilities on the open Millennium Prize Problems. OpenAI said its agents reached their Navier-Stokes resolution on September 5, about 88 hours after the first agents were launched.
The company said the proof was “produced by an internal OpenAI system” and showed “that the dynamics of the Navier-Stokes equations for fluid motion can develop a singularity in finite time.”
“We’re sharing both a writeup of the proof and a formalization in Lean,” OpenAI said.
The company described the 90 year old problem as one of the “deepest questions at the frontier of mathematics”, adding that the question of whether smooth three-dimensional fluid motion can break down “has remained unresolved for roughly 90 years.”
OpenAI said its work was aimed not only at the mathematical problem but also at demonstrating the pace at which artificial intelligence systems are advancing.
“A major goal of our work is to empower scientists to advance research and technology that benefits all of humanity,” the company said.
“To solve the Navier–Stokes problem, we used an internal model that is significantly more capable than GPT-6 Astra. We believe it is important to inform the world about the pace of AI progress and what to expect from upcoming models.”
How the AI approached the problem
OpenAI said the Navier-Stokes equations use Newton’s second law of motion to describe how fluids move and are used in areas including aircraft design, weather forecasting and the study of blood flow.
It said a fundamental question has been whether the equations can develop a “singularity” even when fluid motion begins smoothly.
“Here, a singularity means the dynamics lead to speeds in the fluid growing without bound within a finite amount of time,” OpenAI said.
The company said its system produced “an analytical proof and a Lean formalization that an initially smooth fluid at rest can develop a singularity in a finite time.”
“The fluid has a smooth force applied to it, and its energy remains finite through the entire dynamics, from rest to the formation of the singularity,” it said.
OpenAI described the mathematical solution as a vortex.
“The solution is a vortex, a spinning swirl of fluid, that spirals inward and gets increasingly elongated, like spaghetti,” the company said.
“This central region shrinks while it speeds up in such a way that its energy still stays finite, as required by the laws of physics.”
The company said the technical challenge was to make the equations produce the breakdown through the movement of the fluid itself rather than by introducing an infinite force.
“The technical challenge is for the equations to develop the breakdown through the motion of the fluid itself, rather than, for example, us putting in an infinite force by hand,” OpenAI said.
10,000 AI agents deployed
OpenAI said the effort began after it started training a new internal model on August 28.
“On Tuesday, September 1, we heard rumors that two Millennium Prize problems had been resolved,” it said.
“Inspired by these rumors and by the step change in performance of our internal model, we launched an effort to evaluate it on all open Millennium Prize problems and a few other high-impact problems.”
The company said it used “a system of coordinating agents powered by our internal model.”
“The agents had access to tools such as the ability to read from a cached version of the internet and the ability to run code,” OpenAI said.
“Agents were subdivided into groups with the ability to communicate within the group. The groups varied in size, and the group that produced the Navier–Stokes resolution involved on the order of 10,000 concurrent agents.”
OpenAI said different groups were given different versions of the problem to explore possible proofs and disproofs.
It also said the agents were tested on an easier related problem involving the Euler equations.
“Nearly 100 agents worked together for approximately 50 hours to produce our Euler regularity disproof,” the company said.
After that result, OpenAI said it decided to concentrate its resources on Navier-Stokes.
“Once we saw the Euler solution, we thought that Navier–Stokes was the most promising problem to work on,” it said.
“To do so, we shifted agents away from the other Millennium Problems and prompted these agents with the Euler resolution.”
OpenAI said it encouraged the different groups to pursue diverse approaches before using Codex to consolidate useful insights.
“After some time, we cross-pollinated the agent groups by using Codex to consolidate the most useful insights from each agent group,” it said.
The company said the agents reached their Navier-Stokes resolution on September 5.
“The agents arrived at their resolution on Saturday, September 5, about 88 hours after the first agents were launched,” OpenAI said.
“Lean formalization and verification took an additional 17 hours via GPT-6 Astra.”
The scale of the effort was also substantial. OpenAI said that across all the problems it attempted, its agents sent 4.9 million messages and used about 300 billion output tokens.
For the Navier-Stokes problem specifically, “the agents sent 2.7 million messages and used approximately 130 billion output tokens,” the company said.
OpenAI says it will not claim prize
Despite describing the result as a resolution of the Navier-Stokes Millennium Prize problem, OpenAI said it would not seek the associated prize.
“Our goal in releasing this result is to report on the substantial progress of our AI models,” the company said.
“We do not intend to claim the Millennium Prize for this result.”
OpenAI also described the work as a milestone rather than an endpoint.
“This milestone represents substantial work by mathematicians and AI researchers. However, this is not a culmination, but rather a snapshot in time, of progress on AI development,” it said.
The company added: “We believe we are now in the next period of AI progress, and today’s results provide further evidence of this.”
OpenAI said it would continue studying the internal model and using what it learns to guide future advances in AI capability.
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