# Show there doesn’t exist a non-constant polynomial $p(x)$ with integer coefficients such that $p(x)$ is prime for all non-negative integers $n$ [duplicate]

Show there doesn’t exist a non-constant polynomial $p(x)$ with integer coefficients such that $p(x)$ is prime for all non-negative integers $n$.
There is a hint: note in particular that $p(0)$, the constant term of $p(x)$, is prime.
• Once you know that the constant $q = p(0)$ is a prime, or perhaps $-q$ is the prime, what happens to the value of the polynomial when $x$ is divisible by $q?$ – Will Jagy Sep 25 '17 at 1:28
• Suppose that $p(0)=a\in\Bbb P$. Notice that $p(a)$ is composite iff $p$ is a non-constant polynomial satisfying the constraints in the problem. – Prasun Biswas Sep 25 '17 at 1:29
• Try $x^2 + x + 5.$ What happens when $x$ is divisible by $5?$ – Will Jagy Sep 25 '17 at 1:43