Short, lame co-proof:
Via Galois, equations of degree five and above have no algorithmic solution.
From number theory (I think), every integer can be decomposed into four squares.
Thus, a^n + b^n = c^n becomes
(a1^2 + a2^2 + a3^2 + a4^2)^n
{similarly for the b and c terms}.
Note: if n>2, (expl: 3), the terms inside the parenthesis A1, A2, A3 and A4 all are higher than degree five, running into the Galois thing.