Question 9
We consider all infinite sequences $(x_n)_{n \ge 1}$ of positive integers that satisfy the recurrence
$$
x_{n+2} = \text{mcd}(x_{n+1}, x_n) + 2006,
$$
for all $n = 1, 2, \dots$. Here $\text{mcd}(u, v)$ denotes the greatest common factor of $u$ and $v$.
Can it happen that a sequence of this type contains exactly $10^{2006}$ different numbers?
$$
x_{n+2} = \text{mcd}(x_{n+1}, x_n) + 2006,
$$
for all $n = 1, 2, \dots$. Here $\text{mcd}(u, v)$ denotes the greatest common factor of $u$ and $v$.
Can it happen that a sequence of this type contains exactly $10^{2006}$ different numbers?