Question 13
Let coefficients of the polynomial $P(x) = a_d x^d + \ldots + a_2 x^2 + a_0$ where $d \ge 2$, are positive integers. The sequence $(b_n)$ is defined by $b_1 = a_0$ and $b_{n+1} = P(b_n)$ for $n \ge 1$. Prove that for any $n \ge 2$, there exists a prime number $p$ such that $p \mid b_n$ but it does not divide $b_1, b_2, \ldots, b_{n-1}$.