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.