$x^2 + y^2 = z^2$ vsFermat's last theorem states that there are no non-trivial integer solutions to $x^3 + y^3 = z^3$..$x^n+y^n=z^n$ for $n=3$ (or in fact any $n\geq 3$), but there are of course many Pythagorean triples that give solutions for $n=2$.
