Tristan Buckmaster is a mathematician at New York University. He poses for a portrait in his Manhattan office on September 11, 2026. Buckmaster and his collaborator Levent Alpöge have been working on the Navier-Stokes equations. They are clashing with OpenAI, which claims to have solved the equations. Karsten Moran/for The Washington Post/Getty Images hide caption toggle caption Karsten Moran/for The Washington Post/Getty Images When OpenAI announced that its AI had solved one of the world’s toughest math problems earlier this month, its tone was celebratory: “An important goal of our work is to empower scientists to advance research and technology that benefit all of humanity,” the company wrote. But mathematicians say that, so far, humanity has learned very little from the solution the company’s AI model supposedly found. Although they believe the test is technically correct, the dense 166-page manuscript drafted by AI is proving to be difficult reading. “So far it’s been very difficult to really extract any human understanding from this new AI test,” said James Maynard, a mathematician at the University of Oxford. “The paper is not written for humans,” said Javier Gómez-Serrano, a mathematician at Brown University who uses AI in his own research. He said he thinks the test could help advance the field after “some serious rewrites,” but “as of today, the paper doesn’t teach us much.” The solution came as AI appears to be rapidly acquiring mathematical knowledge. AI has burst onto the mathematics scene in the past six months, said academic mathematicians contacted by NPR. For the first time, large linguistic models appear capable of producing real results that could lead to new mathematical discoveries. But few see the OpenAI announcement, which came as human mathematicians were closing in on a solution, as an example of how AI and mathematicians can work together. “This whole episode could have been a wonderful proof of concept of the power of human-AI collaboration,” said Maynard, who signed a statement from 25 recipients of the prestigious Fields Medal. The statement, which was released on September 11, denounced “misaligned goals” between AI companies and the mathematics community. “Unfortunately, due to the decisions of the humans involved, it became a complicated and messy battle,” he said. Rushing to discovery The problem that OpenAI apparently solved is considered one of the most important unanswered questions in mathematics. Known as the “Navier-Stokes problem”, it is a set of equations used to describe fluid flow. Equations are used every day in physics and engineering, said Tristan Buckmaster, a mathematician at New York University. And yet, at a fundamental level, researchers don’t really understand why they work. Buckmaster says that gaining deeper insights into Navier-Stokes would likely produce “a new set of tools for understanding complicated solutions to fluids.” This, in turn, could lead to better models of things like turbulence in fluids and lift in airplanes. To find a better version of Navier-Stokes, researchers had been looking for scenarios in which the equations “broke” or stopped working. Describing such scenarios became one of the Million Dollar Millennium Prize Problems, which were established in 2000 by the Clay Mathematics Institute. Mathematicians had been working for years to find an answer to the Navier-Stokes problem, and the community could sense that they were getting closer. “Of the seven Millennium Problems… Everyone agreed that this would be the next one to be solved,” said Martin Hairer, a mathematician at EPFL, a leading European technical university located in Lausanne, Switzerland, who also signed the statement. Buckmaster and his collaborator, Anthropic’s Levent Alpöge, were among the human mathematicians who were getting closer to finding a scenario in which the equations stopped working. The two were using AI tools, including the OpenAI chatbot, to work toward a solution. Then OpenAI learned that the problem could be solved soon. “On Tuesday, September 1, we heard rumors that two Millennium Prize issues had been resolved,” the company wrote. “Inspired by these rumors and the radical change in the 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 brought together about 10,000 AI agents to work on the Navier-Stokes problem. The agents worked for 88 hours, using around 130 billion output tokens, or units of text and characters generated by the model. Based on OpenAI’s pricing for its most advanced publicly available model, calculating the solution to the Navier-Stokes problem cost the company between $6 million and $10 million. (Overall, the company says it used 300 billion tokens, closer to $15-20 million, looking for solutions.) The announcement of the OpenAI solution sparked controversy after Buckmaster issued a public statement describing how the company had approached him. According to Buckmaster, they said they would include him in their newspaper if he left Alpöge, his collaborator at the rival company Anthropic. Buckmaster told NPR that he believes OpenAI was informed of the approach he and Alpöge were using to try to find the solution to Navier-Stokes. “There is so much circumstantial evidence that they had a lot more knowledge of what we were doing than they were letting on,” he said. He added that the word “they” does not necessarily mean a person. “It could refer to the agents,” he said. OpenAI has denied using Buckmaster and Alpöge’s guidance or evidence in its search for a solution. Still, the rushed results seem confusing. Several mathematicians who examined the OpenAI paper said it was virtually unreadable. “It’s a terribly written article,” Buckmaster said. “The document doesn’t explain what parts are important. What parts are routine? How is the idea conveyed to other places?” Gómez-Serrano added. Buckmaster said OpenAI’s decision to rush a response forced it to publish its own preliminary results heavily influenced by AI. These too, he said, are not very well written. “Everything was rushed,” he said. “It’s still not at the level that makes me happy, but at least the introduction has all the key ideas.” Computers review computers While OpenAI’s 166-page proof is taking weeks to unravel, few mathematicians believe it is wrong. This is because, in addition to publishing the test, the company also produced computer code called Lean formalization. Lean is a programming language used in mathematics to check whether proofs are correct. If the code can be compiled, then the test is considered complete. The Lean code produced by OpenAI tools compiled as expected, Gómez-Serrano said. Based on that, “the community seems to have consensus that this is correct.” Gomez-Serrano, Buckmaster and other mathematicians contacted by NPR said they envisioned a future for AI in mathematics. The field is open to new tools, in part because for decades it has given up much of the everyday calculations needed to make advances in computers. But this episode is a reminder that math is about more than just finding a solution, Maynard said. “It wasn’t just about responding to this problem, it was about the human understanding behind it,” he said.
