P = NP
Frequently called the most important outstanding question in theoretical computer science, the equivalency of P and NP is one of the seven problems that the Clay Mathematics Institute will give you a million dollars for proving — or disproving. Roughly speaking, P is a set of relatively easy problems, and NP is a set that includes what seem to be very, very hard problems, so P = NP would imply that the apparently hard problems actually have relatively easy solutions. But the details are more complicated.
The Millennium N-Queens Completion Problem
"... has been Solved. 7 NP-Complete & 4 NP-Hard Problems have been solved by us in Polynomial Time and some in Logarithmic Time." - Founders, Automatski