The Millennium Prize Problems are a collection of the most difficult problems in mathematics, including the Riemann Hypothesis and the P versus NP Problem. Of the seven problems, only the Poincaré Conjecture has been solved. The mathematical problem that took the longest to solve was Fermat's Last Theorem.
The Riemann Hypothesis states that all non-trivial zeroes of the Riemann zeta function are complex numbers with real part 1/2. This is said to be useful in determining the distribution of prime numbers.
The P versus NP problem is a conundrum involving the determining of the amount of time to find a solution to a given problem versus the time it takes to verify that solution.
Fermat's Last Theorem states that x^n + y^n = z^n has no non-zero integer solutions for x, y and z when n > 2.
After the death of Pierre de Fermat in 1665, his son went through his papers with the intent of publishing the collected ideas and comments. The theorem was found written in the margin of a published text, along with the note "I have discovered a truly remarkable proof which this margin is too small to contain."
Despite his claim to have proven the theorem around 1630, the problem remained unsolved until 1994 by Andrew Wiles of Princeton University.