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 A_{1}, A_{2}, A_{3} and A_{4} all are higher than degree five, running into the Galois thing.