Representation of integers as powers of the golden ratio

Question

How to prove that any integer $n$ can be represented in the form of $$n= \phi^{z_1}+\phi^{z_2}+\phi^{z_3}+...+\phi^{z_m}$$ For $z_1$, $z_2$... $z_m$ $\in$ $\mathbb Z $ and $\phi =\frac{ \sqrt 5+1}{2} .$

Answer 1

$$1 = \phi^0$$ $$1 = \phi^{-1} + \phi^{-2}$$ Note that $$\phi^{-n} = \phi^{-(n+1)} + \phi^{-(n+2)}$$ You may expand the sum onto any integers.