Question 47
For each integer $n \ge 2$, we consider the set of all $n$-th roots of unity $U_n = \{z \in \mathbb{C} \mid z^n = 1\}$. Determine the integers $n \ge 2$ for which there exist $x, y \in U_n$, $x \ne y$ such that $x + y \notin U_k$ for any integer $k \ge 2$.