Let $f(n)=a_3n^3+a_2n^2+a_1n+a_0$, with $a_i\in\mathbb{Z}$, $a_3>0, a_0\neq 0$ such that $f(n)>0$ for all positive integers $n$.

Given a prime $p$, when is $f(p)$ again prime?

For example, let $f(n)=7n^3-50n+30$. Then, $$f(7)=2081\quad {\rm (prime)},$$ $$f(11)=19\cdot463,$$ $$f(13)=14759\quad {\rm (prime)}.$$

Are there conditions on the $a_i$'s that guarantee that $f(p)$ is prime for all primes $p$?

anypolynomial $f$ of degree greater than 1 whether $f(n)$ is prime for infinitely many values of $n$, prime or otherwise. This is the Buniakowsky conjecture (when $f$ satisfies the obvious necessary condition). $\endgroup$