Millennium Prize Problems: Which Ones Are Solved?

Of the seven Millennium Prize Problems announced in 2000, exactly one has been solved: the Poincaré Conjecture. Grigori Perelman proved it in work published in 2002 and 2003; the other six remain unsolved as of November 2026.

What the Millennium Prize Problems are

The Clay Mathematics Institute (CMI), a private nonprofit mathematics organization, announced the problems on May 24, 2000, at the Collège de France in Paris. The initiative was intended to spotlight important open questions in mathematics and encourage major advances. CMI offered a prize of US$1 million for a qualifying solution to each problem.

These are not ordinary exercises with a known method waiting to be applied. Each problem asks for a proof of a deep mathematical claim, or, in the case of the Yang–Mills problem, a rigorous construction establishing a physical and mathematical theory’s properties. A proposed answer must withstand expert scrutiny and meet CMI’s rules; an announcement alone does not settle a problem.

The seven problems are:

• Birch and Swinnerton-Dyer Conjecture
• Hodge Conjecture
• Navier–Stokes Existence and Smoothness
• P versus NP
• Poincaré Conjecture
• Riemann Hypothesis
• Yang–Mills Existence and Mass Gap

The solved problem: Poincaré Conjecture

The Poincaré Conjecture concerns the shape of three-dimensional spaces, expressed using topology. In simplified terms, it says that a closed, simply connected 3-manifold is homeomorphic to the 3-sphere. “Closed” means compact and without boundary; “simply connected” means every loop in the space can be continuously contracted to a point. A 3-sphere is the three-dimensional analogue of a sphere’s surface, not the ordinary two-dimensional surface of a ball.

Henri Poincaré posed the conjecture in 1904. It became a central question in topology because it links a broad, abstract description of a space—how loops behave—to a precise classification of its overall form. The result is specifically about three dimensions. The corresponding generalized Poincaré conjecture in dimensions other than four was resolved through work by other mathematicians before Perelman’s proof of the three-dimensional case.

Perelman’s achievement built on Richard Hamilton’s Ricci flow program. Ricci flow evolves a manifold’s geometry in a way loosely comparable to how heat spreads, smoothing irregularities. Singularities can form during the flow, so proving the conjecture required a method for understanding and controlling them. Perelman’s papers supplied crucial arguments that completed the geometrization program and, as a consequence, proved the Poincaré Conjecture.

How Perelman’s proof was confirmed

Perelman posted his work on the arXiv preprint server in 2002 and 2003. A preprint is a research paper shared before or outside the standard journal publication process; its appearance does not, by itself, certify that a proof is correct. Mathematicians then studied the papers, checked the difficult details, and published extensive accounts explaining and verifying the arguments.

The verification involved more than finding a short, isolated proof of the conjecture. Perelman established results that yielded Thurston’s Geometrization Conjecture, a broader classification statement about three-dimensional manifolds. The Poincaré Conjecture follows as a special case. Independent expositions by teams of mathematicians helped make the long and technically demanding proof accessible for detailed review.

CMI recognized the proof as a solution under its prize rules in 2010. Perelman declined the prize money, as well as the Fields Medal awarded to him in 2006. Declining an award does not reverse the mathematical result: the conjecture is considered solved because the proof was checked and accepted by the mathematical community.

The six problems still open

The remaining questions span number theory, geometry, fluid mechanics, computer science, and mathematical physics. Their statements are technical, but their central themes can be sketched:

• Birch and Swinnerton-Dyer: Connects the number of rational solutions on an elliptic curve with the behavior of a related L-function at a particular point.
• Hodge Conjecture: Asks whether certain geometric features of complex algebraic varieties can be represented by combinations of algebraic subvarieties.
• Navier–Stokes Existence and Smoothness: Seeks a proof that, under specified conditions in three dimensions, the equations describing fluid flow have globally existing, smooth solutions—or a valid example showing failure.
• P versus NP: Asks whether every problem whose proposed solution can be checked efficiently can also be solved efficiently. It is a foundational question in theoretical computer science.
• Riemann Hypothesis: Predicts that all nontrivial zeros of the Riemann zeta function have real part one-half. It is closely connected to the distribution of prime numbers.
• Yang–Mills Existence and Mass Gap: Requires a rigorous mathematical foundation for quantum Yang–Mills theory and proof of a positive mass gap, a feature associated with the theory’s particle spectrum.

“Unsolved” means that no solution has been accepted under the relevant standards—not that nobody has made progress. Researchers have developed partial results, special cases, computational evidence, and new techniques for these subjects. For example, extensive calculations have tested many zeros relevant to the Riemann Hypothesis, but checking examples cannot establish a claim about infinitely many zeros.

What counts as a solution—and the prize

A compelling argument must address the exact problem as stated, not just a related special case or a large collection of examples. It must be sufficiently clear and complete for experts to inspect. For the Millennium prizes, CMI’s rules also set a process: a proposed solution must be published in a qualifying outlet, at least two years must pass after publication, and it must receive general acceptance in the worldwide mathematics community.

The million-dollar figure is a prize offered for each problem, not a payment that automatically goes to whoever first claims a result. In Perelman’s case, CMI awarded the prize after the proof had been reviewed and accepted; he declined it. The prize program therefore combines a financial award with a demanding verification process.

Common misconceptions

One common confusion is that the seven problems are all “the Millennium Prize Problem.” There are seven separate problems, each with its own statement and prize. Another is that a paper posted online, a numerical confirmation, or a proof of a restricted version automatically solves one. None does without a complete argument and broad expert acceptance.

It is also inaccurate to say that the Poincaré Conjecture remains open because Perelman refused the award. The prize and the theorem are separate matters. The proof’s status rests on its mathematical validity and review, not on whether its author accepted recognition or money.

Frequently Asked Questions

How many Millennium Prize Problems have been solved?
One of the original seven has been solved: the Poincaré Conjecture. The other six remain open.

Who solved the Poincaré Conjecture?
Grigori Perelman proved it in papers posted in 2002 and 2003, using ideas that completed Richard Hamilton’s Ricci flow program.

Did Perelman accept the million-dollar prize?
No. CMI awarded him the prize in 2010, and he declined it. He had also declined the Fields Medal in 2006.

Is P versus NP solved?
No. It remains one of the six open Millennium Prize Problems. No proof that P equals NP or that P differs from NP has been accepted.

Can a problem be solved without its author claiming the prize?
Yes. Prize acceptance is not required for a mathematical result to count as a solution. The proof must satisfy the problem’s standards and gain expert acceptance.

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