Question 80
$$
P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0
$$
with integer coefficients *primitive* if and only if $\text{gcd}(a_n, a_{n-1}, \dots, a_1, a_0) = 1$.
a) Let $P(x)$ be a primitive polynomial with degree less than $1398$ and $S$ be a subset 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 exists a primitive polynomial $P(x)$ with degree less than $1398$ such that for each *natural* number $n$, $P(n)$ is divisible by every prime less than $1398$.
P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0
$$
with integer coefficients *primitive* if and only if $\text{gcd}(a_n, a_{n-1}, \dots, a_1, a_0) = 1$.
a) Let $P(x)$ be a primitive polynomial with degree less than $1398$ and $S$ be a subset 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 exists a primitive polynomial $P(x)$ with degree less than $1398$ such that for each *natural* number $n$, $P(n)$ is divisible by every prime less than $1398$.