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Six of the seven Millennium Prize Problems remain unsolved. The Clay Mathematics Institute (CMI) lists five under “Unsolved” and lists Navier–Stokes separately as “Active”; both labels refer to problems that have not been solved. The sixth remaining problem is Yang–Mills existence and the mass gap.
The six open problems at a glance
| Problem | Field and central question |
|---|---|
| Birch and Swinnerton-Dyer Conjecture | Number theory: how does an elliptic curve’s rational points relate to the behavior of its associated L-function at s = 1? |
| Hodge Conjecture | Algebraic geometry: which topological features of certain algebraic varieties can be represented by algebraic subvarieties? |
| Navier–Stokes existence and smoothness | Fluid equations: under the conditions in the formal problem, do solutions exist and remain unique and smooth, or can a breakdown occur? |
| P versus NP | Theoretical computer science: can every problem whose proposed solution is efficiently checked also be efficiently solved? |
| Riemann Hypothesis | Number theory: do all nontrivial zeros of the Riemann zeta function have real part 1/2? |
| Yang–Mills existence and the mass gap | Mathematical physics: can a rigorous four-dimensional quantum Yang–Mills theory be constructed for compact simple groups with a positive mass gap? |
These are six different mathematical challenges, not variations on one puzzle. Their intuitive summaries are useful entry points, but each official problem has a precise technical statement.
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Birch and Swinnerton-Dyer: rational points on elliptic curves
An elliptic curve is an algebraic object whose rational points—solutions with rational-number coordinates—can be studied using number-theoretic tools. The Birch and Swinnerton-Dyer Conjecture links the number, or rank, of those rational points to the behavior at s = 1 of an associated L-function.
Elliptic curves also have applications, including in cryptography. That connection does not make the conjecture itself a cryptographic product or a practical security procedure: the prize problem is to prove the mathematical relationship.
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Hodge: when topology comes from algebra
The Hodge Conjecture asks whether certain topological features of suitably well-behaved algebraic varieties can be represented by algebraic subvarieties. In broad terms, it concerns whether information that can be described using topology has an algebraic-geometric counterpart.
Some special cases are known. CMI notes, for example, that the conjecture is known when the solution set has dimension less than four; the dimension-four case remains open. Those partial results do not settle the general conjecture.
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Navier–Stokes: existence and smoothness of fluid-flow solutions
The Navier–Stokes equations describe fluid motion, including the flow of water and air. The problem asks, under the conditions laid out in the formal statement, whether solutions exist and are unique and smooth—or whether a counterexample or breakdown can occur.
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This is a question about the mathematical behavior of the equations, not a promise that solving it would immediately produce more accurate weather forecasts or engineering designs. CMI labels this prize problem “Active,” rather than placing it in its “Unsolved” section; it remains open.
P versus NP: checking a solution versus finding one
CMI puts the central question plainly: “If it is easy to check that a solution to a problem is correct, is it also easy to solve the problem?” In complexity theory, “easy” is formalized in terms of efficient computation.
For example, finding a Hamiltonian path—a route that visits each vertex in a graph exactly once and returns to its start—can be difficult, while checking a proposed route is comparatively straightforward. P versus NP asks whether every problem for which a proposed answer can be checked efficiently can also be solved efficiently. The question was formulated independently by Cook and Levin in 1971.
Riemann Hypothesis: the location of zeta-function zeros
The Riemann Hypothesis states that every nontrivial zero of the Riemann zeta function has real part 1/2. The conjecture is connected to the distribution of prime numbers and, in particular, to how their distribution deviates from its average pattern. Bernhard Riemann formulated it in an 1859 paper.
CMI’s Riemann Hypothesis page, accessed in 2026, reports that 10,000,000,000,000 nontrivial zeros had been checked. Checking a finite number of cases computationally is evidence about those cases, not a proof that the statement holds for every nontrivial zero.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Yang–Mills: a rigorous theory with a mass gap
This problem asks for a rigorous construction or existence result for quantum Yang–Mills theory on four-dimensional space for compact simple groups, together with proof of a positive mass gap. It sits at the mathematical foundations of quantum field theory: the challenge is to establish the theory and its required property rigorously.
It is not a request to find a particular particle experimentally. The “mass gap” is part of the mathematical property the constructed theory must satisfy.
Why these seven problems were selected
CMI established the prize series to mark the new millennium and highlight important open questions. The problems were announced in Paris on 24 May 2000. CMI designated a total prize fund of $7 million, allocated as $1 million for each problem.
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What it takes for a solution to earn a prize
A claimed proof does not by itself mean that a Millennium Prize has been awarded. Under rules CMI revised in 2018, the institute does not accept direct submissions. A proposed solution must first meet publication and review conditions before CMI will consider it.
Quick Recap
- Publication: The proposed solution must be published in a qualifying outlet.
- Time: At least two years must pass after publication.
- Acceptance: The solution must receive general acceptance in the global mathematics community.
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