A problem regarding polynomials with prime values

The problem is as follows:

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

I cannot solve it. I can't even find the exact definition of a non-constant polynomial. Any help would be appreciated.

Answer

There are still some gaps, but I'd suggest something like the following. There must be quite a few other approaches, I'd expect, and I hope others will provide some of these.

Suppose a formula exists that produces primes for all positive integers, then $P(1)$ is prime, say $p$. Moreover, $P(1+np)\equiv 0 (\text{mod } p)$ for all natural numbers $n$. Since these values must all be prime, $P(1+np) = p$. There are infinitely many positive integers $n$ and therefore this is only possible if $P(n) = p$ for all $n\in\mathbb{N}$, which is a constant polynomial.