The following paper provides result on an analog of twin primes conjecture for $\Bbb F_q[X]$ http://pollack.uga.edu/twins.pdf

Is there an analog of twin primes conjecture for $\Bbb Z[X]$?

An analog similar to that in the paper of the form "Given $g(X)\in\Bbb Z[X]$, $\forall i\in \Bbb N,\mbox{ }\exists f_i(X)\in\Bbb Z[X]$ with $\operatorname{deg}(f_i)>\operatorname{deg}(g)$ and $\forall i,j\in\Bbb N$, $i\neq j\implies f_i(X)\neq f_j(X)$ such that $f_i(X),f_i(X)+g(X)$ are both irreducible in $\Bbb Z[X]$".