Inverse Fermat's theorem

Question

Wiles proved that Fermat's last theorem is true, but... does it stand for inverse case? Does equation $\frac{1}{x^n}+\frac{1}{y^n}=\frac{1}{z^n}$ have no whole number solutions for $n>2$?

Answer 1

Let's assume that for some $x, y, z\in \mathbb Z\setminus\{0\}$ and $n\in \mathbb N\setminus \{1, 2\} $ the following equation holds $$\bigg(\frac{1}{x}\bigg)^n+\bigg(\frac{1}{y}\bigg)^n=\bigg(\frac{1}{z}\bigg)^n\iff (zy)^n+(zx)^n=(xy)^n$$ The assumption contradicts Fermat's Last Theorem since $zy, zx, xy \in \mathbb Z$ and was therefore wrong $\square$