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Six of the seven Millennium Prize Problems remain unsolved. The Clay Mathematics Institute (CMI) lists five of them under “Unsolved” and labels Navier–Stokes “Active,” meaning it is still an open prize problem. The only one CMI lists as solved is the Poincaré Conjecture. These labels reflect CMI’s problem pages as checked in 2026, so confirm the current listing on CMI’s site before treating them as final.
Status of all seven problems
| Problem | Field | CMI label |
|---|---|---|
| Birch and Swinnerton-Dyer Conjecture | Number theory (elliptic curves and L-functions) | Unsolved |
| Hodge Conjecture | Algebraic geometry | Unsolved |
| Navier–Stokes existence and smoothness | Partial differential equations of fluid flow | Active |
| P versus NP | Theoretical computer science | Unsolved |
| Riemann Hypothesis | Analytic number theory | Unsolved |
| Yang–Mills existence and mass gap | Quantum field theory (mathematical foundations) | Unsolved |
| Poincaré Conjecture | Topology | Solved |
Counting matters here. Five entries sit in CMI’s “Unsolved” section, but six problems are open once Navier–Stokes is included. Its “Active” label does not mean progress toward a solution has been recognised; it means the problem is still open on CMI’s list.
The six open problems
These problems use different tools and sit in different fields, so they are not six versions of the same puzzle. Each entry below gives the core question in CMI’s terms.
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Birch and Swinnerton-Dyer Conjecture
An elliptic curve has a set of rational points, and the conjecture relates the number of those points, or their rank, to the behaviour at s = 1 of an L-function attached to the curve. Elliptic curves matter across number theory, and they also appear in cryptography. The prize, however, is for proving the mathematical conjecture, not for any cryptographic system built on elliptic curves.
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Hodge Conjecture
Roughly stated, it asks which topological features of a suitably well-behaved algebraic variety can be represented by algebraic subvarieties. CMI notes that the conjecture is known in certain special cases, including cases where the dimension is less than four. The dimension-four case remains open.
Navier–Stokes existence and smoothness
The equations describe the motion of fluids such as water and air. CMI’s own phrasing of the question is “do solutions exist, and are they unique?” The formal problem asks whether smooth solutions exist and remain unique under the conditions in CMI’s official description, or whether a breakdown can occur. A solution would be a rigorous mathematical theorem. It would not by itself produce more accurate weather forecasts or engineering designs.
P versus NP
CMI states the question as: “If it is easy to check that a solution to a problem is correct, is it also easy to solve the problem?” Its accessible illustration contrasts finding a Hamiltonian path through a graph with checking a path that someone proposes. Stephen Cook and Leonid Levin formulated the problem independently in 1971.
Riemann Hypothesis
The hypothesis asserts that every nontrivial zero of the Riemann zeta function has real part 1/2. These zeros are closely tied to how the prime numbers deviate from their average distribution. Bernhard Riemann formulated the hypothesis in his 1859 paper. CMI’s Riemann Hypothesis page reports that 10,000,000,000,000 (ten trillion) nontrivial zeros had been checked. That is a finite computational check. It does not prove the statement for all nontrivial zeros.
Yang–Mills existence and mass gap
The problem asks for a rigorous construction or existence result for quantum Yang–Mills theory on four-dimensional space, for compact simple gauge groups, together with a positive mass gap. It is a question about the mathematical foundations of quantum field theory. It does not ask physicists to discover a new particle in an experiment.
Why the list exists
CMI established seven prizes to mark the new millennium and to draw attention to important open questions. The problems were announced in Paris on 24 May 2000. CMI designated a $7 million prize fund, allocated as $1 million for each problem. In its overview, CMI states the aim was “to elevate in the consciousness of the general public the fact that, in mathematics, the frontier is still open and abounds in important unsolved problems.”
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.How a solution earns the prize
CMI revised its prize rules in 2018. It does not accept direct submissions. A proposed solution must meet each of the following conditions before it can be considered:
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- The proof is published in a qualifying outlet, as defined in CMI’s rules.
- At least two years pass after publication.
- The result receives general acceptance in the global mathematics community.
A news story, or an author’s claim to have proved a problem, does not by itself mean a prize has been awarded.
Where to check the current status
For the technical description and status of each problem, use CMI’s individual problem pages. For the award procedure, use CMI’s rules page. The formal statements and prize rules are also collected in the official edited volume The Millennium Prize Problems, which CMI describes as giving the official description of each of the seven problems and the rules governing the prizes.
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