Problem

Source: Iranian third round 2019 Finals Number theory exam problem 2

Tags: number theory



Call a polynomial $P(x)=a_nx^n+a_{n-1}x^{n-1}+\dots a_1x+a_0$ with integer coefficients primitive if and only if $\gcd(a_n,a_{n-1},\dots a_1,a_0) =1$. a)Let $P(x)$ be a primitive polynomial with degree less than $1398$ and $S$ be a set of primes greater than $1398$.Prove that there is a positive integer $n$ so that $P(n)$ is not divisible by any prime in $S$. b)Prove that there exist a primitive polynomial $P(x)$ with degree less than $1398$ so that for any set $S$ of primes less than $1398$ the polynomial $P(x)$ is always divisible by product of elements of $S$.