The Millennium Prize Problems Explained Simply: Seven Questions That Changed Maths

In the year 2000, a private foundation in Massachusetts — the Clay Mathematics Institute — put $1 million on the table for the solution to each of seven mathematical questions. Not easy questions: the kind that the best mathematicians alive had already failed to crack, some for over a century. Twenty-six years later, six are still open, one has been solved, and one more is, as of 2 October 2026, in a genuinely interesting in-between state. Here's what each of the seven actually asks, in plain English, and exactly where each one stands today.

1. The Riemann Hypothesis

The question, in one sentence: is there a hidden, exact pattern behind how the prime numbers (2, 3, 5, 7, 11, 13…) thin out as numbers get bigger?

Primes look scattered and unpredictable one at a time, but there is a special mathematical object — the Riemann zeta function — whose behaviour is tied directly to how primes are distributed. The Riemann Hypothesis predicts that every one of this function's “non-trivial zeros” (specific points where it equals zero) lies on a single, precise line. If that's true, it would pin down exactly how evenly primes are spread out, far more precisely than anything currently proven.

Where the analogy stops: it's tempting to picture this as “finding a pattern in the primes themselves,” but the hypothesis is really a statement about a different, constructed function that encodes information about primes — not a direct rule like “every other prime is…”

Why it matters: huge areas of number theory already assume the Riemann Hypothesis is true and build on top of it — a proof would instantly upgrade hundreds of existing results from “true if RH is true” to simply “true.”

Status: open. Over 10 trillion of these zeros have been checked by computer, and every single one sits exactly on the predicted line. No exception has ever been found — and no proof that one can never exist, either.

2. P versus NP

The question, in one sentence: if a problem's solution is quick to check once you have it, is it also always quick to find in the first place?

Think of a giant sudoku, or finding the best possible delivery route through 500 cities. Checking a completed sudoku, or checking whether a given route is under some target length, takes almost no time. Actually solving it from scratch, as the puzzle gets bigger, seems to take dramatically longer — the number of possibilities explodes. P versus NP asks whether that gap between checking and solving is a fundamental, unavoidable fact of mathematics, or just a reflection of nobody having found a clever enough method yet.

Why it matters: almost every password, every secure website, every piece of encrypted data relies on certain problems being easy to check but hard to solve. If P = NP were proved true — and a genuinely fast method existed — most of modern cryptography would need rebuilding from scratch.

Status: open. The overwhelming majority of computer scientists believe P ≠ NP (that the gap is real and permanent), but nobody has proved it either way.

3. The Navier–Stokes Existence and Smoothness Problem

The question, in one sentence: do the equations that describe how fluids (water, air, anything that flows) move always stay well-behaved forever, or can a perfectly smooth starting flow suddenly “blow up” into infinite speed or turbulence in finite time?

The Navier–Stokes equations are the maths behind weather models, aircraft design, and ocean currents — physicists and engineers use them constantly, with huge practical success. What was never proven is whether the equations themselves are mathematically trustworthy in every case: given smooth, reasonable starting conditions, do they always keep producing smooth, sensible answers forever, or can the maths itself go wrong (a “singularity”) even though nothing physically crazy is actually happening?

Status, as of 2 October 2026: genuinely in-between. In September 2026, a result addressing this problem was announced — reported by several outlets as connected to an OpenAI research effort — proposing a case where a smooth starting flow does blow up in finite time. The Clay Mathematics Institute responded on 11 September 2026 that the problem has “apparently been settled,” while stating plainly that its evaluation process is “deliberately unhurried.” That is not the same as a formal prize award — see the Distinction box below.

4. The Yang–Mills Existence and Mass Gap Problem

The question, in one sentence: can the mathematics underpinning the Standard Model of particle physics be put on fully rigorous footing — and, if so, does it correctly predict that the lightest possible particle it allows for has a strictly positive mass (a “mass gap”), exactly as real experiments observe?

Postgraduate extensionPhysicists already use Yang–Mills theory constantly, and it matches experiments to extraordinary precision. What's missing is a fully rigorous mathematical construction of the theory itself — the kind of airtight foundation mathematicians, rather than physicists, require — plus a proof that the theory genuinely forces this mass gap to exist, rather than it simply being observed in practice.

Why it matters: it sits at the exact junction of physics and pure mathematics — a solution would be one of the deepest confirmations yet that the mathematical structure of modern physics is logically sound, not just empirically successful.

Status: open. Considered by many mathematicians to be among the hardest and most technically demanding of all seven.

5. The Hodge Conjecture

The question, in one sentence: for a certain well-behaved family of shapes built from polynomial equations, do all of their “natural” internal structures actually come from simpler algebraic pieces, as expected?

Postgraduate extensionThis lives in algebraic geometry, the study of shapes defined by polynomial equations (the same family of objects behind elliptic curves, below). Certain shapes have internal features that can be described using topology (the study of shape in a very flexible, stretch-and-bend sense) — the Hodge Conjecture predicts that, for a specific well-behaved class of these shapes, every one of those topological features can also be built directly out of algebraic pieces (sub-shapes that are themselves solutions to polynomial equations). It's a question about whether two different ways of looking at the same shape — topological and algebraic — always agree.

Status: open, and widely regarded as one of the most difficult to even state precisely for a general audience — proven true in many specific cases, but not as a general rule.

6. The Birch and Swinnerton-Dyer Conjecture

The question, in one sentence: for a particular kind of curve (an “elliptic curve”), can you predict how many whole-number or fraction solutions it has, just by looking at the behaviour of a related function?

Elliptic curves are equations of the form y² = x³ + ax + b — deceptively simple-looking, and central to modern cryptography (including the encryption protecting most banking apps) and to Andrew Wiles' famous proof of Fermat's Last Theorem. Some elliptic curves have infinitely many rational-number solutions; others have only finitely many. The conjecture says you can tell which kind you're looking at — and even estimate how “rich” its solutions are — entirely from the behaviour, near a specific point, of a separate function built from that curve (its L-function).

Why it matters: it would turn a currently case-by-case, often computationally difficult question (“how many solutions does this curve have?”) into something you could answer from a clean, provable rule.

Status: open, though verified computationally for a very large number of individual elliptic curves — every curve checked so far behaves exactly as the conjecture predicts.

7. The Poincaré Conjecture — the one that's solved

The question, in one sentence: if a 3-dimensional shape has no holes in it (in a precise topological sense) and is otherwise well-behaved, must it actually be a (higher-dimensional) sphere, just possibly stretched or deformed?

In two dimensions, this is almost obvious: any loop you draw on a sphere's surface can be shrunk down to a point without leaving the surface, but a loop drawn around a doughnut's hole can never shrink to a point while staying on the doughnut's surface — so a sphere and a doughnut are genuinely different shapes. The Poincaré Conjecture asked whether the same “no holes means it's secretly a sphere” rule holds one dimension up, for 3-dimensional shapes living in 4-dimensional space — far harder to picture, and far harder to prove.

Status: solved. Grigori Perelman proved it in 2002–2003, using a technique called Ricci flow (developed by Richard Hamilton) that gradually “smooths out” a shape's geometry, step by step, until its true structure becomes clear.

Summary table

Try it yourself: evidence vs. proof

Several of these problems — Riemann's zeros, Goldbach-style questions, the Collatz conjecture — share the same trap: overwhelming numerical evidence that feels like proof but isn't one. Drag the slider below to see why.

Where to go next

This article covers the seven official Millennium Prize Problems. Mathematics has plenty of other famous open questions outside that list — the Twin Prime Conjecture, the Collatz Conjecture, the abc conjecture and more — each with its own, often very different, status. Our Unresolved Mathematical Mysteries hub covers all of them side by side, with the same care about what's actually proven versus merely suspected. And if you're curious whether AI could solve any of these for us, Can AI Actually Do Maths? looks at exactly that question.