For n = 1, 2, ... , we give a solution (x[subscript 1], ... , x[subscript n], N) to the Diophantine integer equation [image omitted]. Our solution has N of the form n!, in contrast to other solutions in the literature that are extensions of Euler's solution for N, a sum of squares. More generally, for given n and given integer weights m[subscript 1], m[subscript 2], ... , m[subscript n] we give a solution to [image omitted]. The weights may be positive or negative and are subject to some restrictions. Choosing weights plus or minus 1 gives a solution to the problem of finding integer vectors of the same length.