Question 3
Let $(x_n)_{n \ge 0}$ be a sequence of integer numbers that fulfills a linear recursion of order $k$ for a fixed positive integer $k$, i.e., there exists real constant numbers $c_1, c_2, \dots, c_k$ such that $x_{n+k} = \sum_{r=1}^k c_r x_{n+k-r}$, $\forall n \ge 0$. Suppose $k$ is the minimum positive integer with this property. Prove that $c_j \in \mathbb{Z}$, for all $j$, $1 \le j \le k$.