The seven Millennium Prize Problems are major unsolved questions selected by the Clay Mathematics Institute (CMI) in 2000. They cover number theory, computer science, geometry, topology, fluid mechanics and mathematical physics. CMI created a US$7 million fund, with US$1 million assigned to each problem.
One problem, the Poincaré Conjecture, has been solved and awarded. CMI said on September 11, 2026, that the Navier–Stokes problem had “apparently been settled,” but that claim remains under scrutiny and has not yet produced a formal CMI acceptance or prize award. CMI lists the other five as unsolved.
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The seven problems at a glance
| Problem | Field and object | Core question | Status reported by CMI |
|---|---|---|---|
| Birch and Swinnerton-Dyer Conjecture | Number theory; elliptic curves | Can information about an elliptic curve modulo primes determine the rank of its rational points? | Unsolved |
| P versus NP | Theoretical computer science; computational problems | If a proposed answer can be checked quickly, can an answer also be found quickly? | Unsolved |
| Poincaré Conjecture | Topology and geometry; three-manifolds | Is the three-dimensional sphere the unique simply connected three-manifold? | Solved; prize awarded |
| Hodge Conjecture | Algebraic geometry and topology; algebraic varieties | How much topological structure in an algebraic solution set comes from algebraic subvarieties? | Unsolved |
| Riemann Hypothesis | Number theory; the Riemann zeta function | Do all nontrivial zeros of the zeta function lie on the line with real part one-half? | Unsolved |
| Navier–Stokes Equation | Fluid mechanics and partial differential equations | Do smooth, physically meaningful solutions exist for all time in three-dimensional space? | Apparently settled announcement under review as of September 11, 2026 |
| Yang–Mills & the Mass Gap | Mathematical physics; quantum Yang–Mills theory | Can the theory be constructed rigorously and shown to have a positive mass gap? | Unsolved |
These are orientation-level descriptions. CMI’s official volume, The Millennium Prize Problems, contains the formal statements and the prize rules.
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Why these problems were created
CMI announced the problems in 2000 to spotlight mathematical questions that had resisted sustained work and to recognize breakthroughs of historic importance. The seven were chosen from different areas rather than as variations on one theme. Together they ask about the arithmetic of prime numbers and elliptic curves, the limits of algorithms, the structure of geometric spaces, the behavior of fluids, and the foundations of quantum field theory.
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The prize fund is US$7 million in total. The official allocation is US$1 million for each problem, so a solution to one problem does not divide a single jackpot among all seven. A claim of a proof and an awarded Millennium Prize are separate events: CMI applies a formal review process before recognizing a solution.
1. Birch and Swinnerton-Dyer Conjecture
What it studies
An elliptic curve is an equation of a particular cubic form, but the prize problem concerns more than drawing the curve. Mathematicians study the rational points—solutions whose coordinates are fractions—and organize those points into a group. The group has a rank, a measure of how many independent rational points generate the rest.
The conjecture in plain language
The Birch and Swinnerton-Dyer Conjecture proposes a link between that rank and data obtained by reducing the curve modulo many prime numbers. The relevant information is encoded in an analytic function associated with the curve: the behavior of the function near a particular point should reveal the rank of the rational-point group.
Why it matters
The conjecture sits at a crossroads of arithmetic geometry and the study of L-functions. CMI notes connections with Fermat’s Last Theorem, prime factorization and cryptography. Those connections do not mean that solving the conjecture would automatically break or create a cryptographic system; they show why understanding elliptic curves has consequences beyond a single theorem.
2. P versus NP
The distinction between finding and checking
In complexity theory, class P contains problems that a conventional deterministic computer can solve in polynomial time, meaning the resources grow in a manageable mathematical way as the input gets larger. Class NP contains problems for which a proposed solution can be checked in polynomial time.
The central question
P versus NP asks whether every problem whose answer can be checked efficiently can also be solved efficiently. CMI’s standard illustration is a Hamiltonian path: given a proposed route that visits each city under the required rules, checking it is straightforward; discovering such a route in an arbitrary network may be far harder.
Practical significance and limits
A proof that P equals NP would show that efficient algorithms exist for every efficiently verifiable problem in NP, including many problems used in scheduling, optimization and security. A proof that P differs from NP would establish a fundamental limit on that approach. Neither outcome, by itself, would provide a ready-to-run algorithm or identify the exact practical difficulty of every real-world instance.
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What the conjecture asked
The conjecture concerns three-dimensional manifolds: spaces that locally resemble ordinary three-dimensional space but may have a complicated global shape. It asked whether a closed, simply connected three-manifold must be the three-dimensional sphere. “Simply connected” means every loop can be continuously contracted to a point.
Perelman’s proof and recognition
Grigori Perelman posted preprints in 2002 and 2003 that developed a proof using Ricci flow and established the broader Thurston Geometrization Conjecture, of which the Poincaré Conjecture is a special case. After the work was checked by mathematicians, CMI announced the Millennium Prize award in 2010.
This is the only one of the seven problems for which CMI has completed the prize process and announced an award. The result is not merely a computer verification of many examples; it is a general mathematical proof covering every space that meets the hypotheses.
4. Hodge Conjecture
Algebraic equations and hidden topology
Take a system of polynomial equations and consider its set of complex solutions. That set can have rich topological features—cycles and holes—that can be studied with tools from algebraic topology. Algebraic geometry also supplies special subspaces cut out by polynomial equations, called algebraic subvarieties.
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The Hodge Conjecture asks how much of the relevant topological information can be represented as combinations of these algebraically defined subvarieties. It predicts that certain classes identified by Hodge theory are algebraic in origin, rather than merely topological artifacts.
What is known
CMI reports that the conjecture is proved in some special situations, including cases where the solution set has dimension less than four. The dimension-four case remains unknown. Partial results therefore do not settle the full statement: the prize requires a theorem that handles the general scope of the conjecture.
5. Riemann Hypothesis
The zeta function and prime numbers
The Riemann zeta function is initially defined by an infinite series and then extended through complex analysis. Its zeros encode information about how prime numbers are distributed. The “trivial” zeros are understood; the difficult ones are the nontrivial zeros in the critical strip.
The claim
The Riemann Hypothesis says every nontrivial zero has real part one-half. In geometric language, the zeros should all lie on a single vertical line in the complex plane. That location would give exceptionally precise control over the irregularities in the distribution of primes.
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CMI reports that the first 10,000,000,000,000 solutions (zeros) had been checked computationally. Such a calculation can test an enormous range and expose no counterexample there, but it cannot establish that every zero, including infinitely many beyond the checked range, lies on the critical line. The conjecture remains listed as unsolved.
6. Navier–Stokes Equation
The mathematical problem
The Navier–Stokes equations model the motion of fluids such as water and air. The Millennium problem is not simply to solve the equations for one engineering setup. It asks whether, in three-dimensional Euclidean space, smooth initial conditions always lead to smooth solutions for all time—or whether a solution can develop a singularity in finite time.
Why smoothness is difficult
Nonlinear terms describe how the fluid transports and stretches its own motion. In three dimensions, those interactions could in principle concentrate energy into smaller scales and cause quantities such as velocity or vorticity to become unbounded. Mathematicians have strong results for restricted cases, but the general existence-and-smoothness question has resisted proof.
The September 2026 status
On September 11, 2026, CMI responded to an announcement that the Navier–Stokes problem had “apparently been settled.” CMI said the work would be analysed and interrogated and that its process was deliberately unhurried. The statement did not announce formal acceptance, identify a prize recipient or award the US$1 million prize.
Accordingly, “apparently settled” is the accurate status for that dated announcement. It should not be rewritten as “solved” until CMI’s review satisfies its rules and an award is announced.
7. Yang–Mills & the Mass Gap
From physics to rigorous mathematics
Yang–Mills theory is a foundation of the Standard Model of particle physics. Physicists use it successfully to describe fundamental interactions, but the mathematical formulation requires more than matching experimental predictions.
The two-part challenge
The problem asks for a rigorous construction of a quantum Yang–Mills theory and a proof that it has a positive mass gap. A mass gap means that the vacuum is separated from the lightest possible excitation by a nonzero amount of energy; mathematically, the theory should not contain arbitrarily low-energy excitations above the vacuum.
Evidence versus proof
Experiments and computer simulations suggest that a mass gap exists. CMI emphasizes that no proof of the required property is known. Numerical evidence can guide conjectures and test approximations, but the Millennium problem demands a mathematically rigorous construction and theorem.
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Which problems are solved?
CMI’s overview identifies the Poincaré Conjecture as solved and the other five—Birch and Swinnerton-Dyer, P versus NP, Hodge, Riemann and Yang–Mills—as unsolved. Navier–Stokes occupies a distinct, time-sensitive category: CMI acknowledged a reported settlement on September 11, 2026, while keeping the work under review.
The distinction matters because a proposed proof is not automatically a prize-winning solution. A claim must pass mathematical scrutiny and CMI’s formal recognition process.
How CMI recognizes a solution
CMI’s revised rules, adopted September 26, 2018, set conditions before it considers a proposed solution:
- The work must be published in a qualifying publication outlet; CMI does not accept direct submissions.
- At least two years must pass after publication.
- The solution must receive general acceptance in the global mathematical community.
Only after those conditions and CMI’s own evaluation can the institute consider awarding the corresponding US$1 million prize. This procedure explains why a headline announcing a claimed proof can precede an official prize decision by years.
How the seven differ
The problems are often grouped together because of the prize, not because they use the same methods. Their objects and tests are fundamentally different:
- Arithmetic structure: Birch and Swinnerton-Dyer and the Riemann Hypothesis connect abstract functions with prime-related information.
- Computation: P versus NP asks about the resources required to find and verify solutions.
- Geometric structure: Poincaré and Hodge ask when global shape can be characterized by algebraic or topological conditions.
- Continuous dynamics: Navier–Stokes asks whether a nonlinear physical model remains mathematically regular.
- Quantum theory: Yang–Mills requires both a rigorous field-theory construction and a nonzero spectral gap.
Comparing them by field, object, question type and status prevents a common mistake: treating a checked computation, a physical simulation, a proposed proof and a formally awarded prize as equivalent kinds of evidence.
Where to find the official statements
CMI’s official overview provides the current list and high-level status. Its individual pages describe the Riemann Hypothesis, Poincaré Conjecture and Navier–Stokes problem in more detail, while the institute’s prize rules define recognition requirements. The edited volume The Millennium Prize Problems by James Carlson, Arthur Jaffe and Andrew Wiles collects the official problem descriptions and rules and is listed by CMI as available through the American Mathematical Society bookstore.
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