Prove that there is no non-constant polynomial $p(n)$ with integer coefficients that only takes prime values **Attempt (so far):** Assume there exists a non-constant polynomial $p: \mathbb Z \to \mathbb Z$ with integer coeffecients that only takes on prime values. Let notate it as $$p(n)=d_j n^j + d_{j-1} n^{j-1}+ \dots +d_1 n + d_0 $$ where $j\in \mathbb N$. Let $k$ be some composite number with $r$-many factors. Then $$\begin{array} \ k &=& p(n_1)^{i_1} p(n_2)^{i_2} \dots p(n_r)^{i_r} \\\ &=& (d_j {n_1}^j + d_{j-1} {n_1}^{j-1}+ \dots +d_1 {n_1}+d_0)^{i_1} \dots (d_j {n_r}^j + d_{j-1} {n_r}^{j-1}+ \dots +d_1 {n_r}+d_0)^{i_r} \end{array}$$ * * * I don't really see how to progress from here unless I want to start doing ungodly amounts of computation. Could someone provide a hint of a path I should be taking?

Given $p(n)=d_j n^j + d_{j-1} n^{j-1}+ \dots +d_1 n + d_0$ note that $p(0) = d_0$ is prime. So then perhaps $d_0$ would be a prime dividing $p(d_0)$, which is also prime. When can one prime divide another? Also note that this happens for any $p(kd_0), k \in \mathbb Z$. How many times can a non-constant polynomial revisit the same value?