The Millennium Prize Questions

The Millennium Prize Questions are seven unanswered questions in mathematics that were stated by the Clay Mathematics Institute (CMI) in 2000. These are some of the most difficult questions in the field of mathematics. A correct answer to any one of these questions carries a reward of $1 million.

Here are the seven Millennium Prize Questions:

1. The P vs NP Question: Is every problem whose solution can be verified in polynomial time (class P) also a problem whose solution can be found in polynomial time (class NP)?

2. The Hodge Conjecture: Is it possible to count the number of independent cycles of a particular type on algebraic varieties?

3. The Birch and Swinnerton-Dyer Conjecture: How can the distribution of rational solutions to equations defining an elliptic curve be determined?

4. The Navier–Stokes Existence and Smoothness Question: Do solutions always exist (and are they smooth) for a specific set of differential equations that model the flow of incompressible fluids?

5. The Riemann Hypothesis: Do all non-trivial zeros of the Riemann zeta function $\zeta(s)$ lie on the critical line $\Re(s) = 1/2$ in the complex plane?

6. The Yang–Mills Existence and Mass Gap Question: Can the existence of quantum field theories corresponding to the standard model of particle physics be established?

7. The Poincaré Conjecture: Can three-dimensional spheres be characterized among all three-dimensional spaces?

The Clay Mathematics Institute (CMI) offers a $1 million prize for each answered question to inspire researchers to tackle these challenging mathematical questions. As of my knowledge cutoff in September 2021, only the Poincaré Conjecture has been definitively answered.