Explained with graphics: The epic math problem OpenAI claims to have solved

5 min readNew DelhiUpdated: Sep 15, 2026 09:03 AM IST

OpenAI said its internal AI system has produced a proof showing that the dynamics of the Navier-Stokes equations for fluid motion can develop a singularity in finite time. (Express Image/OpenAI)

OpenAI said Tuesday that its artificial intelligence (AI) model had solved one of the most popular and unanswered problems in mathematics – the Navier-Stokes existence and smoothness problem.

One of the seven Millennium Prize Problems – a group of major math questions identified by the Clay Mathematics Institute in 2000, each of which has a $1 million prize for a recognised solution – the Navier-Stokes question deals with whether smooth three-dimensional fluid motion can break down, and it has remained unsolved for about 90 years.

OpenAI said its internal AI system has produced a proof showing that the dynamics of the Navier-Stokes equations for fluid motion can develop a singularity in finite time. But what is the problem, and why does it matter?

Decoding Navier-Stokes equations

Imagine trying to predict exactly how water in a river will move over time.

This is what the Navier-Stokes equations do; they are essentially a set of mathematical rules that explain how any fluid (water, honey, air, etc.,) flows based on two simple physics ideas, such as fluid mass and force equals mass times acceleration.

These equations can help in forecasting weather, designing aeroplanes and cars, and predicting ocean currents.

The catch here is nobody has proven whether the equations always produce sensible answers.

Story continues below this ad

Since fluid flow is chaotic, the difficulty comes from turbulence. This means tiny changes in starting conditions could lead to wildly different outcomes, which is also why weather cannot be predicted more than a week ahead.

What has OpenAI found?

AI generated explainer

OpenAI also said that the proof has been formally verified using Lean, which is a computer system used to check mathematical proofs step-by-step.