Most Interesting Open Math Questions

1. Twin Prime Conjecture: There are an infinite number of twin primes. (This problem was known to the ancient Greeks; it’s related to Hilbert’s Eighth Problem.)

2. Goldbach Conjecture: Every even integer larger than 2 is the sum of two primes. (This problem was known to the ancient Greeks; it’s related to Hilbert’s Eighth Problem.)

3. Riemann Hypothesis/Conjecture: All of the non-trivial zeros of the Zeta Function are located on the line Re(s) = ½. (This is the central point of Hilbert’s Eighth Problem; also one of the Millennium Problems.)

4. Birch & Swinnerton-Dyer Conjecture: If a given elliptic curve has an infinite number of solutions, then the associated L-series has value 0 at a certain fixed point. (One of the Millennium Problems)

5. Hodge Conjecture: For projective algebraic varieties, Hodge cycles are actually rational linear combinations of algebraic cycles. (One of the Millennium Problems)

6. Jacobian Conjecture: If det[F’(x)]=1 for a polynomial map F, then F is bijective with polynomial inverse. (One of Steve Smale’s 1998 problems)

7. Under what conditions does a solution exist to the Navier-Stokes non-linear partial differential equation? Do these equations have solutions that last for all time, given arbitrary sufficiently nice initial data, or do singularities develop in the fluid flow that prevent the solution from continuing? (One of the Millennium Problems)

8. How many limit cycles are possible for a given ODE? (Half of Hilbert’s Sixteenth Problem)

9. Describe the shapes possible for the graphs of algebraic functions with only real numbers allowed as solutions. (Half of Hilbert’s Sixteenth Problem)

10. Does P = NP? (One of the Millennium Problems)

The Poincare Conjecture is not included in this list because based on what I've read (not that I'm qualified to evaluate a lot of what's being written about it) it seems that the solution by Perelman is correct.

A plethora of additional unsolved mathematics problems can be found at mathworld.wolfram.com.