$x\in \mathbb{R}\setminus\{0\}$ and $x + \frac{1}{x}$ is an integer. Prove $\forall n\in \mathbb{N}$ $x^n + \frac{1}{x^n}$ is also an integer.
Proving that $\frac{\phi^{400}+1}{\phi^{200}}$ is an integer.