OpenAI has announced that it discovered a solution to a longstanding advanced mathematics problem in a matter of hours, utilizing a new artificial intelligence model and approximately 10,000 autonomous AI agents.
The problem pertains to the Navier-Stokes equations, which describe the movement of fluids. For 90 years, crucial aspects of these equations have lacked a formal proof, which is the foundational argument supporting a correct mathematical equation.
OpenAI described the discovery as a “milestone,” citing it as evidence of the rapid advancement of AI tools.
The company noted that at the end of August, it initiated training for a new model that rapidly demonstrated exceptional mathematical proficiency. While these AI models are computer programs trained on vast datasets to identify and predict patterns, this particular model remains an internal tool that is significantly more capable than OpenAI’s most recent public release. Researchers chose to apply it to prominent advanced mathematics problems.
On September 1, OpenAI admitted it had “heard rumors that two Millennium Prize problems had been resolved.” Consequently, the company deployed thousands of AI bots trained on the new internal model to attempt solutions to the remaining problems.
By September 5—approximately 88 hours after assigning 10,000 AI bots to the task—OpenAI had found a solution to the Navier–Stokes existence and smoothness problem.
This problem lies at the core of turbulence, a phenomenon that remains poorly understood.
Although the AI bots seemingly required very little time to reach a solution, OpenAI stated that they exchanged nearly 3 million messages and consumed 130 billion output tokens—the individual lines of text and code generated by the AI—solely for the Navier–Stokes problem.
Based on OpenAI’s own pricing for output from its most advanced models, this level of effort would have cost roughly $10 million (£7.3 million).
OpenAI stated that the solution it reached resolves two out of the four statements required in the proof demanded by the Millennium Prize. The prize is awarded to the first to solve certain mathematical conundrums and is worth $1 million to the winner.
“Our goal in releasing this result is to report on the substantial progress of our AI models,” OpenAI stated on Tuesday. “We do not intend to claim the Millennium Prize for this result.”
However, the company’s claim has already sparked controversy.
Tristan Buckmaster, a mathematics professor at New York University, stated on Tuesday that he and Levent Alpöge, a mathematician working for OpenAI rival Anthropic, had also been developing solutions for the problem. The duo had been utilizing OpenAI’s tool, Codex. However, Buckmaster noted that on September 3, he discovered that “information about our progress had been passed to OpenAI.”
Buckmaster’s statement came the same day, but hours before OpenAI published its Navier–Stokes work. He alleged that OpenAI did not begin working on the Navier–Stokes equations until after receiving information about their work. He included text from emails exchanged with OpenAI regarding the work and questioned the company’s timing and methods.
Buckmaster added that he had not yet read OpenAI’s full proof, but felt compelled to go public with “what I was told, when, and what was proposed to me… because the alternative is to let a sequence of announcements say something I know to be false.”
On Tuesday, OpenAI congratulated the “concurrent work” of Buckmaster and Alpöge, calling it “remarkable.”
The company stated it had not seen “any of their work through any means until they released it publicly” and that no user data was accessed during its work on the Navier–Stokes problem.
“While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models,” the company added. “However, our proofs differ significantly and even the precise results proved are different.”
