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

  • $\begingroup$ A minor point is that, in your Problem 2 statement, I assume that $d$ is also an integer, but you should make that explicit. $\endgroup$ – John Omielan Dec 25 '18 at 4:17
  • $\begingroup$ $d$ is integer, edited! $\endgroup$ – Drona Dec 25 '18 at 4:25
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    $\begingroup$ @Drona can you post your solution for p=2 and 4? $\endgroup$ – Sandeep Silwal Dec 25 '18 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 $m$.

  • $\begingroup$ Thanks Robert !! $\endgroup$ – Drona Dec 25 '18 at 5:49

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