Millennium Prize Problems

In the year 2000, a private research institute in the USA, the Clay Mathematics Institute, presented a list of seven mathematical questions. All seven were unsolved at the time, and some had been for over a hundred years. A million US dollars was promised for each complete solution. Since then, these seven problems have been known as the Millennium Prize Problems. To this day, only one of them is considered solved: the so-called Poincaré conjecture, a question about the shape of curved spaces. The Russian mathematician Grigori Perelman proved it around 2003 — and declined the prize money.

Seven questions as a map of open mathematics

The list was never intended as a quiz. It was meant to show where mathematics reaches its limits. Each of the seven questions does not stand on its own, but is tied to an entire field of research. Whoever solves one usually also changes the understanding of hundreds of related problems.

For the tech world, one problem in particular is of interest: the question of whether P equals NP. Put simply, it concerns whether every task whose solution can be quickly verified can also be quickly found. A Sudoku can be checked in seconds, but solving it can take a long time. If someone proved that both can be done equally fast, it would be an earthquake. Many encryption methods on the internet rely on certain calculations remaining practically unsolvable.

Most experts, however, suspect the opposite: that P and NP are different. But a proof of this is likewise missing. This is exactly what makes the list so persistent — it is not about a lack of computing power, but a lack of ideas.

The rules of the prize money

You cannot win the money by sending an email to the institute. A solution must first appear in a respected academic journal. There, other mathematicians examine it step by step. After that, the proof must hold up in the academic community for two years. Only then does the institute appoint a committee that decides on the payout.

This procedure sounds cumbersome, but there is a reason for it. Every year, alleged proofs surface that, upon closer examination, contain gaps. A single flawed step renders an entire proof worthless. With Perelman, it even happened the other way around: he published his work only on an open online archive for academic texts, without a journal. Other teams needed years to fully trace through the arguments.

The remaining six problems concern areas such as prime numbers, the flow of fluids, and the mathematics behind particle physics. A well-known example is the Navier-Stokes equations. They describe how water or air moves, and are used daily in weather models. Yet no one knows whether their solutions always remain well-behaved or whether they might eventually diverge to infinity.

Why AI companies use the list as a benchmark

The term now frequently appears in news about artificial intelligence. When a company presents a new language model, it is often asked whether it might one day solve a Millennium Problem. The list serves as a deliberately extreme goal in this context. It marks the boundary between computation and genuine mathematical insight.

Realistically, AI systems are far from this. They now achieve good results on tasks from mathematics competitions for school and university students. However, such tasks have a known solution and a clear framework. A Millennium Problem requires new concepts and methods that no one has formulated so far.

A common misconception is that these problems are pure brain teasers without any use. The opposite is true. The Riemann hypothesis, for instance, concerns the distribution of prime numbers, and prime numbers are the foundation of many encryption methods. Even where no direct benefit is apparent, tools emerge along the way to a solution that later transform entirely different fields.