OpenAI says it cracked 90-year-old maths problem in 88 hours

OpenAI says it solved a 90-year-old maths problem in 88 hours

OpenAI says it has come up with a solution to a decades-old advanced maths problem in just hours, using a new artificial intelligence (AI) model and thousands of AI bots.

For 90 years, key parts of the problem have lacked a proof, the reasoning behind a correct mathematical equation.

OpenAI described the result as a "milestone" and as proof that AI tools are advancing rapidly.

The company’s solution has not yet been independently verified or publicly accepted by The Clay Mathematics Institute, a US-based maths organization that oversees the Millennium Prize, which offers large sums of money to the first person to solve certain mathematical puzzles.

OpenAI said that at the end of August it began training a new model that quickly proved highly capable at maths.

Although the new OpenAI model is still used only inside the company, and is "significantly more capable" than its latest public AI model release, researchers chose to apply it to some notable advanced maths problems.

OpenAI said that on 1 September last week it had "heard rumors that two Millennium Prize problems had been resolved" and therefore decided to deploy thousands of AI bots trained on the new internal model to try to solve some of the remaining problems.

By Sept 5, or about 88 hours after it had assigned 10,000 AI bots to the task, OpenAI said it had found a solution to the Navier–Stokes existence and smoothness problem.

That problem lies at the core of turbulence, a phenomenon that is still not well understood.

Even though the AI bots appeared to reach a solution quickly, OpenAI said they exchanged nearly 3 million messages and used 130 billion output tokens, meaning the individual lines of text and code an AI model generates in responses, on Navier–Stokes alone.

Based on OpenAI’s own pricing, that effort would have cost about $10m (£7.3m).

OpenAI said the solution it has now reached for the Navier–Stokes existence and smoothness problem settled two of the four statements required in the proof demanded by the Millennium Prize. The prize is worth $1m to the winner.

"Our goal in releasing this result is to report on the substantial progress of our AI models", OpenAI said on Tuesday. "We do not intend to claim the Millennium Prize for this result."

The company’s claim is already drawing controversy.

Tristan Buckmaster, a mathematics professor at New York University, said on Tuesday, external that he and Levent Alpöge, a mathematician working for OpenAI rival Anthropic, had also been pursuing solutions to the problem.

The pair had been using OpenAI’s Codex tool in their work. But Buckmaster said that on 3 September he learned that "information about our progress had been passed to OpenAI."

Buckmaster’s remarks came on the same day, but hours before, OpenAI published its Navier–Stokes work. He said OpenAI did not start working on the Navier–Stokes equations until "after information about our work had reached OpenAI." He shared text from emails exchanged with OpenAI about the work and his concerns over the company’s timing and methods.

Buckmaster also said he had not yet read OpenAI’s full proof, but felt he had to speak publicly about "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 said it had not seen "any of their work through any means until they released it publicly" and that no user data was accessed in 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."