# Inverse Fermat's theorem

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 $$n>2$$?

• What could you say about the triple $(yz,zx,xy)$? – Mindlack Feb 13 '19 at 11:20

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 $$\square$$