$R=\mathbb Z[\sqrt3]. x=2-sqrt3, then {x^n:n is an integer} is an infinite set of distinct values.

Question

Let $R=\mathbb Z[\sqrt3]$

I would like to show that when $x=2-\sqrt3$, then $\{x^n:n \in \mathbb Z\}$ is an infinite set of distinct values. How should I do this?

Thank you!

Answer 1

If for some $n\in \mathbb Z$ $\{(2-\sqrt3)^n:n\in \mathbb Z\}$ does not have distinct values then

$(2-\sqrt3)^i=(2-\sqrt3)^j;1\leq i,j\leq n$

then since $2-\sqrt3$ is invertible in $\mathbb Z[\sqrt 3]$ so $(2-\sqrt3)^{i-j}=1$

which is true only for $i=j$