<s>[INST] Find the values of $n$ such that $(1+\sqrt3i)^n$ is a real number 
Find the values of $n$ such that $z^n=(1+\sqrt3i)^n$ is a real number.

My reasoning: The power will be real iff $\sin\arg z=0$. Since $\sin 0,\sin\pm\pi,\sin\pm2\pi,\dots=0$, $3\mid n$. Is it correct?
$$z=re^{ia}=2e^{i\pi/3}$$
$$z^n=2^n\left(\cos\frac{n\pi}3+i\sin\frac{n\pi}3\right)$$
$$\to\{n\mid n=3k, k\in\Bbb Z\}$$
 [/INST] An alternate view for $a_{n} = (1+ \sqrt{3} i)^{n}$ is:
\begin{align}
a_{0} &= 1 \\
a_{1} &= 1 + \sqrt{3} i \\
a_{2} &= -2 + 2 \sqrt{3} i \\
a_{3} &= -8.  
\end{align}
It is determined that $n \in {0, 3, \cdots}$. Consider $n \to 3n$ then
\begin{align}
a_{3n} &= (1 + \sqrt{3} i)^{3n} = [(1 + \sqrt{3} i)^{3}]^{n} = (-8)^n = (-1)^{n} \, 2^{3n}.
\end{align}
This yields the same result as that given by the proposer's solution.
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