Exercise 1.6 [Goldbach 1752] of An introduction to number theory

Question

Prove that if $f\in \mathbb Z[x]$ has the property that $f(n)$ is prime for all $n\ge 1$, then $f$ must be a constant.

This is a Exercise of An introduction to number theory. In fact, I am not sure what is $\mathbb Z[x]$. Is it integer value function ? Besides, how to show it ?

Answer 1

Hint Let $f(1)=p$.

Show that for each $k$ you have $$p | f(kp+1)-f(1)$$

Deduce that $f(kp+1)$ is divisible by $p$, and since $f(kp+1)$ is prime...

Answer 2

The notation $\mathbb{Z}[X]$ stands for the set of all polynomials with coefficients in $\mathbb{Z}$.