TITLE: Primes of the form $p_{i_1}p_{i_2}\cdots p_{i_n}+2k$
QUESTION [5 upvotes]: Let $S_{n,k}$ be the set of all numbers that can be written as the product of $n$ odd primes plus $2k$. Is there integers $n>1$ and $k>1$ such that $S_{n,k}$ contains finite number of primes?

REPLY [3 votes]: Assuming Schinzel's Hypothesis H, these sets are always infinite:
Step 1:  Let $t$ be the product of $n-1$ primes such that $\gcd(t,2k)=1$.
Step 2:  Let $f(x)=x$ and $g(x)=tx+2k$.  Define $Q(x)=x(tx+2k)$.
Hypothesis H asserts that if $Q(x)$ has no fixed prime divisor $q$, then $f(x)$ and $g(x)$ are simultaneously prime an infinite number of times.
Step 3:  If $x=1$ then $q$ divides $Q(x)=t+2k$, and thus $q$ is odd.  If $x=2$,  then $q$ divides $Q(x)=2(2t+2k)$, and thus $q$ divides $t+k$.
Step 4:  Since $q$ divides both $t+2k$ and $t+k$, we find $q$ divides both $k$ and $t$, contradicting that $\gcd(t,2k)=1$.
Hence $Q(x)$ has no fixed prime divisor, and Hypothesis H applies.  Thus, since $g(x)$ has the desired form, we conclude that $|S_{k,n}|=\infty$.