Question 146
Let $f(x)$ be a polynomial with integer coefficients, which has no multiple roots and $\deg f \ge 1$. Prove that for every positive integer $k$,
$$
P_k(f) = |\{p \in \mathbb{P} \mid \mathrm{ord}_p(f(x)) = k, \exists x \in \mathbb{Z}\}| = \infty.
$$
(proposed by G. Batzaya)
$$
P_k(f) = |\{p \in \mathbb{P} \mid \mathrm{ord}_p(f(x)) = k, \exists x \in \mathbb{Z}\}| = \infty.
$$
(proposed by G. Batzaya)