Proving that $\frac{\phi^{400}+1}{\phi^{200}}$ is an integer. How do we prove that $\dfrac{\phi^{400}+1}{\phi^{200}}$ is an integer, where $\phi$ is the golden ratio?
This appeared in an answer to a question I asked previously, but I do not see how to prove this..

Using the golden ratio $\phi=\frac{1+\sqrt{5}}{2}$ and silver ratio $\psi=\frac{1-\sqrt{5}}{2}$, we get:
$$\dfrac{\phi^{400}+1}{\phi^{200}}=\phi^{200}+\frac{1}{\phi^{200}}=\phi^{200}+\psi^{200}.$$
The Fibonacci formula:
$$\begin{align}F_n&=\frac{\phi^n-\psi^n}{\sqrt{5}} \Rightarrow \\
\phi^{100}-\psi^{100}&=\sqrt{5}\cdot F_{100} \Rightarrow \\
\phi^{200}+\psi^{200}&=5F_{100}^2+2.\end{align}$$
Note: $F_{100}=354224848179261915075$.
