Exercise 5.4.13 (Equation $x^n + y^n = z^n$ if $n \equiv 0 \pmod 4$)

Proof. We write $n=4m$, where $m$ is a positive integer. If Fermat’s equation $x^n+y^n=z^n$ had a solution in positive integers $x,y,z$, then

This shows that the equation $X^4+Y^4=Z^4$ has a solution $(X,Y,Z)=(x^m,y^m,z^m)$ in positive integers. This is impossible by Theorem 5.10.

The Fermat’s equation $x^n+y^n=z^n$ has no solution in positive integers $x,y,z,$ if $n$ is a positive integer, $n\equiv 0(\mathit{mod}4)$. □