Millennium Problem

In 2000, the Clay Mathematics Institute in the USA published a list of seven mathematical questions. All seven had been open for decades, even though many of the best experts had worked on them. The institute set a prize of one million US dollars for each question. Since then, these seven problems have been known as the Millennium Problems. To this day, only one is considered solved: the Poincaré conjecture, a problem about the shape of spaces. The Russian mathematician Grigori Perelman solved it around 2003 and declined the prize money.

What a million dollars reveals about difficulty

The prize money is not really the point. What matters more is that an independent institution used it to publicly establish which questions are considered especially important. The list acts like a compass: it shows young researchers where the great open construction sites lie. And it makes mathematics more tangible for outsiders, because there is a concrete subject one can talk about.

For the tech world, one of the seven problems is especially relevant: the question of whether P equals NP. Roughly speaking, it concerns whether every task whose solution can be checked quickly can also be solved quickly. A Sudoku is a good example: a completed solution can be checked in a minute, but solving it takes considerably longer. If P equaled NP, there would always be a fast procedure for such tasks.

The consequences would be enormous. A large part of today’s encryption relies on the fact that certain computational tasks take a practically unmanageable amount of time to solve. If this assumption fell, online banking in its current form would be insecure. Most experts, however, suspect that P and NP are not equal — but no one has been able to prove it.

How a proof gets recognized

A solution does not consist of a computational result but of a proof. A proof is a seamless chain of argument that leads from known statements to the new statement. Every step must be clear enough that experts can verify it. A single flawed step renders the whole thing worthless, no matter how convincing the rest sounds.

The Clay Institute has fixed rules for this. The work must appear in a respected academic journal and be peer-reviewed there. After that follows a two-year waiting period during which the scientific community examines the result. Only then can the prize money be awarded. This hurdle exists because alleged proofs regularly surface that turn out to be wrong within a few weeks.

A common misconception is that computers could solve such problems if they were only powerful enough. That is not the case. Computing power provides examples and clues, but not a universally valid justification. Proof-assistant software and AI systems today help verify individual steps, but the decisive idea still comes from humans.

From headlines to cryptocurrency

In the news, the Millennium Problems usually come up in two situations. First, when someone claims to have solved one — this happens regularly, for instance, with the Riemann hypothesis about the distribution of prime numbers. Second, when AI companies talk about the capabilities of their systems. Some cite unsolved mathematical problems as a benchmark for when an AI can truly conduct independent research.

The topic also comes up in the world of finance. Anyone discussing the security of Bitcoin or bank encryption will sooner or later arrive at the P-versus-NP question. It is the silent assumption on which these systems are built. Anyone who knows the term understands such debates much more quickly.

The remaining five open problems concern areas such as fluid flow or the physics of elementary particles. They are harder to explain in a single sentence, but are considered equally significant. What all seven have in common is that they do not merely fill a gap in knowledge. Solving each one would transform an entire field of research.