OpenAI announced on Tuesday that its newest artificial intelligence technology had solved one of the 'Millennium Problems', a collection of important unanswered math questions meant to push the world's leading mathematicians to new heights. According to reports from Business Standard, the company's latest model needed just 88 hours to solve what mathematicians call 'the Navier–Stokes existence and smoothness problem'. This problem involves a series of equations that are often used to predict the weather and describes the movement of water and other fluids, asking whether these equations completely break down in certain situations. The breakthrough is the latest to demonstrate how advanced artificial intelligence models consuming enormous amounts of computing power are encroaching on and reshaping the field of mathematics. The solution shows that the Navier-Stokes equations, which describe how fluids flow, can sometimes 'blow up' - creating a so-called singularity where infinitesimally small parts of the fluid begin to flow infinitely quickly. If the result holds up to further scrutiny, it is by a significant margin the most important mathematical proof to have been arrived at by an artificial-intelligence model to date, possibly marking a fundamental turning point in how mathematicians tackle difficult problems.
OpenAI deployed as many as 10,000 'AI agents' working in concert to solve the Navier-Stokes problem, as reported by Business Standard. Running such a large number of AI systems is likely to have cost millions of dollars, a result of the enormous amounts of electrical power needed to operate the specialized chips that drive AI technologies. OpenAI research scientist Noam Brown called this a 'very expensive process,' before adding that the costs of running AI technologies tend to drop as companies like OpenAI improve their efficiency. The company's internal system that tackled the problem was more powerful than its latest GPT-6 Astra model, with OpenAI's GPT-6 Astra taking an additional 17 hours to verify the solution. According to recent reports, OpenAI put an unprecedented amount of resources on this single task, first answering a simplified version in 50 hours using 1,000 AI agents before increasing compute power to solve the full problem. Sébastien Bubeck of OpenAI estimates the computational cost at several million dollars, with the agents sending almost 5 million messages to each other during the process. The effort began on September 1st after hearing rumors about potential Millennium Prize solutions, with the agents working together for approximately 50 hours before the solution was achieved on Saturday, September 5th.
The Navier-Stokes problem was one of seven 'Millennium Problems' selected by the Clay Mathematics Institute in 2000 as a way of tracking the progress of mathematics in the new millennium. The institute, founded by an American businessman named Landon T. Clay, offered a million dollars for the first correct solution to each problem. Before OpenAI's announcement, only one of the problems had been solved previously. According to Business Standard, OpenAI announced that it focused its resources on the Navier-Stokes problem after hearing that other mathematicians were exploring similar research, though the company said it 'did not see any of their work through any means'. The announcement has stirred controversy after mathematician Tristan Buckmaster, a professor at New York University, claimed that OpenAI accelerated its work after hearing rumors that two Millennium Prize Problems had been solved. Recent developments show that on September 7, Buckmaster and Harvard's Levent Alpöge released a paper claiming to have found a solution for the fluid equations that also achieved infinite speed, though in the simplified case where the fluid has no viscosity. The controversy deepened when Buckmaster raised concerns about potential data access, stating in a document posted on his website: 'I do not know what their model did, or how. I do not know whether our data was used. I am not accusing anyone of anything.'
OpenAI's solution used reinforcement learning, a technique where AI systems learn additional behavior through extensive trial and error by working through thousands of math problems. As reported by Business Standard, researchers inside labs like OpenAI develop complex feedback mechanisms that show the system when it has done something right and when it has done something wrong. The company released a paper describing its solution, including a Lean proof, allowing outside mathematicians to see how its agents cracked the problem. OpenAI researcher Sébastien Bubeck described this as 'a spectacular culmination of the arc we have seen over the last 12 months'. The solution to the Navier-Stokes problem suggests that the equations describing fluid movement occasionally 'blow up' with the speed of fluids becoming impossibly infinite. Recent reports indicate that OpenAI computer scientist Ven Chandrasekaran explained that their proof shows 'there exist fluids which start out perfectly normal, and under the Navier-Stokes equations, actually achieve infinite speed in a finite amount of time', noting this behavior is physically impossible for real fluids. The Navier-Stokes equations rely on the assumption that you can zoom in on a fluid, considering endlessly smaller amounts of it, but the real world is not like this - fluids are ultimately made of molecules and atoms that are not perfectly smooth. The solution is described as a vortex, a spinning swirl of fluid, that spirals inward and gets increasingly elongated, like spaghetti, with the central region shrinking while it speeds up in such a way that its energy still stays finite.
The announcement has excited some mathematicians while stirring concern among others about the future of the field. According to Business Standard, Terence Tao, a professor at the University of California, Los Angeles, who is regarded by many as the finest mathematician of his generation, warned that AI technologies could end up damaging the field of mathematics. He compared the situation to 'having machines that can lift weights for you at the gym' rather than human mathematicians getting value from solving problems through effort and understanding. The controversy deepened when Buckmaster raised concerns about potential data access, stating in a document posted on his website: 'I do not know what their model did, or how. I do not know whether our data was used. I am not accusing anyone of anything.' At a press briefing, OpenAI researcher Sebastien Bubeck denied using the pair's work or accessing material shared with OpenAI's servers, though the company acknowledged it 'could not rule out that data from the pair's use of their products helped improve our models.' Recent developments show that mathematician Martin Bridson, president of the Clay Mathematics Institute, described the work by Alpöge and Buckmaster as a 'remarkable achievement' on social media platform Mastodon. The OpenAI announcement came just 12 hours after an announcement from Tristan Buckmaster at New York University that he, together with Levent Alpöge at Anthropic, had resolved several closely related problems with help from AI models. Both AI-enabled teams relied heavily on work by Diego Córdoba of the Institute for Mathematical Sciences in Madrid and Luis Martínez-Zoroa of CUNEF University, who had developed a strategy to attack the problem that radically departed from traditional mathematician methods.
