Suppose $f(x)$ is an irreducible polynomial over $\mathbb Z$ of degree $n$. Is it always the case that there exist distinct $x_1,\ldots,x_{2n+1}\in \mathbb Z$ such that $f(x_1),\ldots,f(x_{2n+1})$ are all prime?
So far I've tested a few irreducible polynomials using mathematica, and this has held. The motivation is to use this as a test for irreducibility, as if $f(x)$ is not irreducible then it takes on at most $2n$ prime values, as either factor takes on $1$ or $-1$ at most $2n$ times. (I'd also be interested in learning if this bound can be improved).
This could potentially be expended to other rings of integers, i.e. $\mathbb Z[i]$, but in that case the minimal number of primes needed to certify that $f(x)$ is irreducible would increase as more units appear.