If a, b, c are all integers greater than 2 then xa+yb = zc has no integer solutions where x, y, z have no common factor greater than 1.
Setting a = b = c gives us Fermat's Last Theorem, showing this new problem to be more general. And probably much harder. It is already known that there can only be a finite number n of solutions to this equation for any given a, b, c. We need to show that n = 0.
The problem was put up initially with US$5000 on offer for a solution, and this amount will be increased by US$5000 for every year that it remains open, till the prize reaches US$50000. If you think you have a solution to this, send it to Professor Daniel Mauldin at the University of North Texas, for he is the head of the Panel that judges potential solutions. If you don't think you'll even dream of considering attempting this problem, you're in good company!