Q:

Proof of Generalized Primorial Primes

Let's call the numbers of the form $k\times p\# \mp1$, the Generalized Primorial Primes.
One can find many $k$ for a fixed $p$ such that $k\times p\# \mp1$ be prime. As an example for $p = 8933$ there are 13 $k$ below 20000. 
I'd like to know, is there any way to prove (or even approach the proof) that there are infinite number of $k$ for each $p$?

$3347\times 8933\#+1$ 
$5570\times 8933\#+1$
$6980\times 8933\#+1$
$7227\times 8933\#+1$
$10087\times 8933\#+1$ 
$11731\times 8933\#+1$
$14532\times 8933\#+1$
$16957\times 8933\#+1$
$17173\times 8933\#+1$
$17575\times 8933\#+1$
$18563\times 8933\#+1$ 
$19518\times 8933\#+1$
$19625\times 8933\#+1$

A:

This is a corollary of Dirichlet's theorem on primes in arithmetic progressions, which states that if $\gcd(a,d)=1$, then the arithmetic progression
$$a,\quad a+d,\quad a+2d,\quad\ldots$$
contains infinitely many prime numbers. For any prime number $p$, letting $d=p\#$  and $a=1$, we see that there are infinitely many prime numbers of the form
$$k\cdot (p\#) +1.$$
For the minus case, just use $a=p\#-1$ instead.

