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The seven Millennium Prize Problems are exceptionally difficult 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, assigning US$1 million to each problem. One problem—the Poincaré Conjecture—is formally solved. CMI lists five as unsolved, while its September 11, 2026 statement says the Navier–Stokes problem has “apparently been settled” but remains under the institute’s review process, with no prize award announced.
The seven problems at a glance
| Problem | Field | What it studies | Question in brief | CMI status |
|---|---|---|---|---|
| Birch and Swinnerton-Dyer Conjecture | Number theory | Elliptic curves and rational points | Can information from an elliptic curve modulo primes reveal the rank of its rational points? | Unsolved |
| P versus NP | Theoretical computer science | Computational complexity | Does every solution that can be checked efficiently also have a solution that can be found efficiently? | Unsolved |
| Poincaré Conjecture | Topology and geometry | Three-dimensional manifolds | Is the three-dimensional sphere the unique simply connected three-manifold? | Solved; prize awarded |
| Hodge Conjecture | Algebraic geometry and topology | Algebraic varieties and their topology | How much topological structure from algebraic equations comes from algebraic subvarieties? | Unsolved |
| Riemann Hypothesis | Number theory | Zeros of the Riemann zeta function | Do all nontrivial zeros have real part one-half? | Unsolved |
| Navier–Stokes Equation | Fluid mechanics and partial differential equations | Three-dimensional fluid flow | Do smooth, physically meaningful solutions always exist in three-dimensional space? | Apparently settled announcement under CMI review |
| 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 |
The status column uses CMI’s published descriptions. “Apparently settled” is not the same as a formally accepted proof or an awarded prize.
Why CMI created the prizes
At the start of the 21st century, CMI identified seven problems whose solutions would represent major advances across mathematics. The institute announced a total fund of US$7 million in 2000 and divided it equally: US$1 million for each problem. The list is deliberately broad. It includes questions about prime-number patterns, algorithms, geometric shape, equations that model fluids and the mathematical foundations of quantum fields.
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A problem can be famous without being close to a solution. Some have attracted generations of partial results, special cases and computer experiments, but none of those achievements automatically resolves the general statement. The prize framework is intended to recognize a complete, durable solution that other mathematicians can examine.
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1. Birch and Swinnerton-Dyer Conjecture
The mathematical object
An elliptic curve is a type of algebraic curve that can be written in a particular cubic form. Of special interest are its rational points—points whose coordinates are ratios of integers. Those points form a group, and the group’s rank measures how many independent rational points generate the rest.
What the conjecture says
The Birch and Swinnerton-Dyer Conjecture proposes a precise link between that rank and the behavior of an associated L-function at a particular point. One way to obtain evidence about the curve is to examine it modulo many prime numbers, where the arithmetic becomes finite and computationally manageable. The conjecture predicts that the pattern seen across those primes reflects the number of independent rational points over the rationals.
Why it matters
Elliptic curves sit at the intersection of pure and applied number theory. They played an important role in the proof of Fermat’s Last Theorem and are used in modern cryptography. A proof would clarify a central structure in arithmetic geometry and strengthen the theoretical basis for methods that analyze elliptic curves.
2. P versus NP
Finding versus checking
Complexity theory classifies computational tasks by the resources needed as the input grows. “P” denotes problems for which a general algorithm can find an answer in polynomial time, a practical benchmark for efficient computation. “NP” denotes problems for which a proposed answer can be checked in polynomial time.
The key question is whether those classes are actually the same: P = NP? If they are, every efficiently checkable solution would also have an efficient discovery method. If they are not, some problems would remain intrinsically hard to solve even though a candidate answer could be verified quickly.
The route-planning example
Suppose a traveler must visit a set of cities without repeating one and wants a route satisfying a specified condition. Checking a supplied route is relatively quick: follow it and verify each requirement. Finding such a route from scratch may require considering an enormous number of possibilities. This contrast illustrates the distinction without proving anything about the full classes P and NP.
Consequences of either answer
A proof that P equals NP would reorganize algorithm design and optimization, because many tasks now treated as computationally intractable would have efficient algorithms in principle. A proof that P does not equal NP would formally establish a fundamental limit on efficient computation. The conjecture remains unsolved; no accepted proof has established either equality or separation.
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The question
Topology studies properties that remain unchanged when a shape is stretched or bent without tearing. In three dimensions, a manifold is a space that locally resembles ordinary three-dimensional space. The Poincaré Conjecture asks whether every closed, simply connected three-manifold is topologically equivalent to the three-dimensional sphere. “Simply connected” means that every loop can be continuously shrunk to a point.
Perelman’s proof and recognition
Grigori Perelman released preprints in 2002 and 2003 that proved the conjecture as a special case of William Thurston’s broader Geometrization Conjecture. The work was checked and developed by the mathematical community. CMI announced the Millennium Prize award in 2010. Perelman declined the prize and other honors, but the mathematical result is regarded as established.
Why it changed geometry
The proof used the Ricci-flow program associated with Richard Hamilton and supplied a classification framework for three-dimensional spaces. It did more than answer a single yes-or-no question: it confirmed the larger geometrization picture, giving topologists a way to understand the possible geometric structures of three-manifolds.
4. Hodge Conjecture
Algebraic equations viewed as spaces
When several polynomial equations are solved simultaneously, their solutions form a geometric object called an algebraic variety. Such a variety can be studied in two complementary ways. Algebraic geometry focuses on the equations, while topology examines features such as holes and cycles that persist under continuous deformation.
The central claim
The Hodge Conjecture asks how much of the topological information identified by Hodge theory can be represented by algebraic subvarieties—geometric pieces that themselves arise from polynomial equations. It predicts that certain rational cohomology classes are combinations of classes coming from these algebraic pieces.
What is known
CMI reports that the conjecture is proved in some special cases, including varieties of dimension less than four. The dimension-four case is still unknown, and the general statement remains open. The difficulty is not finding isolated examples; it is proving that the algebraic description always captures the relevant topological structure.
5. Riemann Hypothesis
Zeta zeros and prime numbers
The Riemann zeta function is defined initially by an infinite series and then extended through complex analysis. Its zeros are complex numbers where the function equals zero. The “trivial” zeros are understood; the important question concerns the nontrivial zeros.
The Riemann Hypothesis states that every nontrivial zero has real part exactly one-half, placing all of them on a vertical line in the complex plane. The location of these zeros controls how regularly prime numbers are distributed. A proof would sharpen many estimates in analytic number theory; a counterexample would force mathematicians to revise a large body of expectations about primes.
What computation can—and cannot—show
CMI reports that the first 10,000,000,000,000 solutions (zeros) had been checked computationally. Checking that many cases is powerful evidence, but it is not a proof that every zero, including ones beyond the tested range, lies on the critical line. The conjecture is therefore still listed as unsolved.
6. Navier–Stokes Equation
The physical equations
The Navier–Stokes equations model the motion of fluids such as water and air. They express conservation of mass and momentum while accounting for viscosity. Engineers and scientists solve versions of these equations numerically every day, but numerical success for particular conditions does not settle the mathematical question posed by CMI.
The Millennium challenge
The prize problem concerns three-dimensional Euclidean space. It asks whether suitable initial conditions always produce solutions that exist for all time and remain smooth, or whether a solution can develop a singularity—a point where quantities such as velocity or its derivatives become unbounded or cease to be well behaved.
The September 2026 status
On September 11, 2026, CMI responded to an announcement that the Navier–Stokes problem had “apparently been settled.” The institute said the work would be analyzed and interrogated and that it would provide updates. CMI described the process as deliberately unhurried. That statement does not announce formal acceptance, does not identify a prize recipient and does not establish that the US$1 million award has been made. Until CMI’s conditions are met, the safest description is an apparent settlement under review.
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From physics to rigorous mathematics
Yang–Mills theory is a foundation of the modern description of fundamental forces. Physicists use it successfully, but the Millennium problem asks for a mathematically rigorous construction of a quantum Yang–Mills theory in four-dimensional space-time.
What “mass gap” means
A mass gap would mean that the vacuum state has zero energy while every nonzero physical excitation has energy at least some positive amount above it. In particle-physics language, the theory would contain no arbitrarily low-energy excitations. Experiments and computer simulations suggest that such a gap exists, but CMI states that no proof of the required property is known.
Why the problem is hard
Quantum field theory combines infinitely many degrees of freedom with quantum mechanics and relativity. Physicists can calculate highly accurate predictions using formal methods, yet turning those methods into a complete mathematical construction—and proving the gap—requires control over the theory at every scale.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Which problems are solved?
CMI formally recognizes the Poincaré Conjecture as solved and awarded its Millennium Prize in 2010. The other five problems—Birch and Swinnerton-Dyer, P versus NP, Hodge, Riemann and Yang–Mills—are listed as unsolved. Navier–Stokes occupies a separate, provisional category because CMI’s September 2026 announcement refers to an apparent settlement while its review is continuing.
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A reported proof, a paper accepted by experts and a Millennium Prize award are three different milestones. A headline or preprint can describe a claim; it cannot by itself change CMI’s official status.
How a claimed solution qualifies for a Millennium Prize
CMI’s revised rules, adopted September 26, 2018, set conditions before the institute considers a proposed solution:
- The work must be published in a qualifying outlet.
- At least two years must pass after publication.
- The result must receive general acceptance in the global mathematics community.
CMI does not accept direct submissions. These requirements give mathematicians time to inspect the argument, identify gaps, reproduce key steps and determine whether the result solves the exact official problem rather than a restricted version. Only after that process can CMI consider awarding the corresponding US$1 million prize.
How the seven problems differ
The list is not seven variations of one theme. The Riemann Hypothesis and Birch–Swinnerton-Dyer Conjecture both concern number theory, but the former concerns complex zeros tied to prime distribution, while the latter concerns rational points on elliptic curves. P versus NP concerns the limits of algorithms rather than a geometric object. Hodge and Poincaré ask about spaces, but Hodge connects algebraic equations to topology and Poincaré classifies a specific kind of three-dimensional manifold. Navier–Stokes asks whether differential equations governing fluids remain smooth. Yang–Mills asks for a rigorous quantum field theory and a positive energy separation.
The status differences are equally important. One problem has a completed proof and award, five remain open, and one has a high-profile claim that CMI has not yet certified.
Where to read the formal statements
Plain-language summaries are useful for orientation, but they omit technical hypotheses and definitions. CMI’s official volume, The Millennium Prize Problems, edited by James Carlson, Arthur Jaffe and Andrew Wiles, contains the formal descriptions and prize rules. Readers who want to assess a proposed solution should use those statements rather than relying on a news headline or an informal explanation.
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