Twin primes for polynomials in $\Bbb Z[X]$ 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 $\Bbb Z[X]$".
 A: Even much stronger statements are nearly immediate consequence of Hilbert's irreducibility theorem. For example: Choose any $f_1$, $f_2$ coprime of $g$, then by Hilbert's irreducibility theorem there exist infinitely many $a\in \mathbb{Z}$ such that $$g(X)+a f_1(X) \quad and \quad g(X)+a f_2(X) $$ 
are both irreducible. Similarly there exist infinitely many $a\in \mathbb{Z}$ such that 
$$
(a f_1)^2 + 1 
$$ 
is irreducible (this is an analogue of a problem of Landau asking whether there are infinitely many primes of the form $n^2+1$).
More generally, a Shinzel Hypothesis H version for $\mathbb{Z}[X]$ is nearly immediate from Hilbert's irreducibility theorem.
Edit: Hilbert's irreducibility theorem asserts that for every $F_1(T,X),\ldots, F_r(T,X)\in \mathbb{Q}(T)[X]$ that are irreducible, there exist infinitely many $t\in \mathbb{Z}$ such that all $F_i(t,X)$ are (defined) and irreducible in $\mathbb{Q}[X]$.
A: Use the Chinese remainder theorem to construct infinitely many $f_i$ such that $f_i$ is Eisenstein at one prime, and $f_i+g$ is Eisenstein at another.
