Prove that there is no non-constant polynomial $P(x)$ such that $P(n)$ is a prime number for all positive integers n

Question

Prove that there is no non-constant polynomial $P(x)$ such that $P(n)$ is a prime number for all positive integers $n$. Appeared in an undergrad entrance exam. I have no idea how to proceed

Answer 1

Suppose $P(x)$ satisfies this property. Let $p=P(1)$.

We know that $p\mid P(kp+1)$ for all $k\in\mathbb N$, so $P(kp+1)=p$ for all $k\in\mathbb N\cup\{0\}$. If you aren't convinced, note that $(kp+1)^i\equiv1\pmod p$.

This is impossible since for any non-constant polynomial, the preimage of a given value must have finite cardinality (consider the polynomial minus the value, that would have infinite roots otherwise), but this implies that $$\{kp+1\mid k\in\mathbb N\cup\{0\}\}\subset P^{-1}(p)$$