# Polynomials have smallest degree

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 $$m\nmid P(n),\quad\forall\,n\in\mathbb N^*.$$

• $d$ is integer, edited! Commented Dec 25, 2018 at 4:25
• @Drona can you post your solution for p=2 and 4? Commented Dec 25, 2018 at 5:17

It seems this is a corollary of the Chebotarev density theorem. See e.g. these notes, Ex. 7.2(b): Let $$f \in \mathbb Z[X]$$ be an irreducible polynomial with the property that $$(f \mod p)$$ has a zero in $$\mathbb F_p$$ for all but finitely many primes $$p$$. Prove that $$f$$ has degree $$1$$.
In particular, in your problem 2 there are infinitely many primes that work for $$m$$.