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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 $\Bbb Z[X]$".

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  • $\begingroup$ Your question is rather vague. What sort of analogue are you looking for? $\endgroup$
    – S. Carnahan ♦
    Commented Nov 27, 2014 at 11:47
  • $\begingroup$ Something 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 $deg(f_i)>deg(g)$ 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 irreducible in $\Bbb Z[X]$". $\endgroup$
    – Turbo
    Commented Nov 27, 2014 at 11:53
  • $\begingroup$ More than just existence of infinitely many twin primes in $\mathbb{Z}[X]$, I'd be interested to know the asymptotics for such prime pairs. That sounds a lot harder, though. $\endgroup$
    – Daniel Hast
    Commented Nov 27, 2014 at 15:24
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    $\begingroup$ @DanielHast Most polynomials of content one in $\mathbb{Z}[x]$ and fixed degree are irreducible, so it's even true that given $g_1,\ldots,g_m$ arbitrary, for most $f$, say monic, of fixed degree $> \max \deg g_i$, $f+g_i$ are all irreducible. $\endgroup$
    – Felipe Voloch
    Commented Nov 27, 2014 at 19:49

2 Answers 2

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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.

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  • $\begingroup$ can we do the same for given two $h,g$ construct infinite examples of form $f_i$, $f_i+g$ and $hf_i+g$ are all irreducible? $\endgroup$
    – Turbo
    Commented Nov 27, 2014 at 20:16
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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 $\mathbb{Q}[X]$.

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  • $\begingroup$ can we replace any $a\in\Bbb Z$ by fixed $a\in \Bbb Z[X]$? $\endgroup$
    – Turbo
    Commented Nov 27, 2014 at 20:17
  • $\begingroup$ I mean fix $g,h$ then are there infinite $f$ of degree bigger than $deg(g)deg(h)$ such that $f,f+g$ and $hf+g$ are all irreducible? $\endgroup$
    – Turbo
    Commented Nov 27, 2014 at 20:20
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    $\begingroup$ You can just replace f_1 by any other f_1 $\endgroup$
    – Lior Bary-Soroker
    Commented Nov 28, 2014 at 7:55
  • $\begingroup$ for completeness could you provide the theorem as well? $\endgroup$
    – Turbo
    Commented Nov 28, 2014 at 10:45
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    $\begingroup$ I added a formulation of HIT $\endgroup$
    – Lior Bary-Soroker
    Commented Nov 29, 2014 at 14:05

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