The 7 Millennium Prize Problems, Explained
Seven questions that have gone unanswered for more than two decades, a million dollars on each one, and a story with more drama than you'd expect.
In May 2000, at a conference at the Collège de France, the Clay Mathematics Institute announced a list of seven unsolved mathematical problems and put one million dollars on the table for a rigorous proof — or disproof — of each. It wasn't the first time someone had put a price on a mathematical problem, but it was the first at this scale and with this level of media attention.
More than two decades later, only one of the seven is officially considered solved. This article introduces all seven, assuming no advanced mathematical background. If you want to go deeper into the one that was solved, we have a dedicated article on the Poincaré conjecture; and if you're curious about the most recent news on one of the still-open problems, we've also written about OpenAI's claimed breakthrough on Navier-Stokes.
Why do these seven problems exist?
The list was explicitly inspired by the 23 problems David Hilbert proposed in 1900, which shaped much of 20th-century mathematical research. The Clay Institute wanted to repeat that gesture, but with a financial incentive and a clear selection criterion: problems that were old, deep, and whose consequences reach far beyond the question itself.
"These aren't puzzles. They're questions that, if answered, change what we know about numbers, shapes, fluids, or the very limits of what a computer can calculate."
The seven problems
1. The Poincaré Conjecture
Asks whether every "hole-free," finite three-dimensional shape is, in essence, a sphere. It sounds intuitive, but proving it rigorously took nearly a century. Grigori Perelman achieved it in 2002-2003 using a technique called Ricci flow with surgery. It's the only one of the seven officially solved and certified.
2. P versus NP
Asks whether every problem whose solution can be checked quickly can also be found quickly. It's the problem with the most practical impact of the seven: the answer determines, among other things, whether there's a mathematical shortcut that could break much of today's cryptography.
3. The Riemann Hypothesis
Concerns where certain special numbers (the "non-trivial zeros" of the Riemann zeta function) are located, and by extension, how prime numbers are distributed. It's arguably the most famous open problem in pure mathematics, and the one most mathematicians would call "the most important" on the list.
4. The Hodge Conjecture
The hardest to explain without formulas: it proposes that certain complicated geometric objects (complex algebraic varieties) can be understood by combining simpler pieces of a purely algebraic nature. It connects geometry, algebra and topology.
5. Yang-Mills Existence and Mass Gap
Comes from particle physics. The Yang-Mills equations describe fundamental forces and work extraordinarily well in practice, but no one has proved with the rigour mathematics demands that their solutions exist consistently, or why the associated particles have mass.
6. Navier-Stokes Existence and Smoothness
The Navier-Stokes equations describe how fluids move (water, air, blood). They're used daily in engineering and meteorology, but it isn't known whether their solutions always exist and stay "smooth," or whether, in theory, a fluid perfectly described by these equations could develop infinite behaviour in finite time. In September 2026 OpenAI announced a possible breakthrough on this problem; we explain exactly what has and hasn't been proven.
7. The Birch and Swinnerton-Dyer Conjecture
Relates the behaviour of a function associated with an elliptic curve to the number of rational solutions of that curve. It sounds very specific, but elliptic curves are at the heart of much of modern cryptography and number theory.
Is the million-dollar prize still up for grabs?
Yes, for the six problems that remain open. The process for claiming it is deliberately demanding: the proof must be published in a prestigious mathematical journal, wait two years for the community to review it, and finally be validated by an expert committee appointed by the Clay Institute. That's exactly what happened with Perelman: he published his results in 2002-2003, and the institute didn't confirm the proof until 2010.
That two-year window matters for understanding any headline claiming a Millennium Prize problem has been "solved": a lot of time usually passes between the announcement and official confirmation, and in some cases — as we'll see — the mathematical community doesn't even agree on whether the announcement actually resolves what the problem asks.
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