I am interested in the following problem.
Problem 1. Find polynomial P (x) has integer coefficients, there is no rational roots, with the smallest degree such that for each positive integer $m$ there exists a positive integer a such that $$m \mid P (a). $$ In the case of $\deg P = 2$, use quadratic residue, we can see that it cannot happen.
In the case of $\deg P = 4$, I have the answer (use quadratic residue).
Therefore, I need an answer for the case of $\deg P = 3$, ie the following problem.
Problem 2. Let the polynomial $ P (x) = ax ^ 3 + bx ^ 2 + cx + d$, $a,\,b,\,c,\,d\in\mathbb Z$ and $a \ne 0 $, $ P (x) $ is irreducible on $ \mathbb Z [x] $. Prove that there exists a positive integer $ m $ such that $$m\nmid P(n),\quad\forall\,n\in\mathbb N^*.$$