Let $z=1+2i$ be a complex number.

Prove that for any $n \in \mathbb{N}^\*$, the number $z^n$ has the following form: $a\_n+ib\_n$, with $a\_n,b\_n \in \mathbb{Z}$.

I guess the solution lies in the trigonometric form of $z$. However, I get a really ugly number trying to transform $z$ into its trigonometric form.

I would really appreciate if I would get some help on this problem.