I have been playing around with this off and on for several years.
Unless I did my algebra wrong, I am now coming up with the following "simple" proof. Anyone see an error here?
1) It can be shown algebraically that (x + y) divides (x^n + y^n) for odd integer n
2) Assume integers x, y, z ( x< y < z) exist such that x^n + y^n = z^n for odd interger n
3) Assume y = x + a and z = x + b. Since x < y < z, b > a > 0
4) From (1) and using (2) and (3) to substitute for y and z , (2x + a) divides (x + b)^n
5) For (4) to be true, the remainder of [(x+b)^n] / (2x +a) must be 0. It can be shown algebraically that the remainder is [b - (a/2)]^n, so this implies b = a/2
6) (5) contradicts (3) that b > a. This implies (2) cannot be true.
Any insights here would be appreciated.