Primes of the form $p_{i_1}p_{i_2}\cdots p_{i_n}+2k$

Question

Let $S_{n,k}$ be the set of all numbers that can be written as the product of $n$ odd primes plus $2k$. Are there integers $n>1$ and $k>1$ such that $S_{n,k}$ contains finite number of primes?

Answer 1

The answer to you question is very likely, no, these sets are always infinite. However, this is open. If you slightly but significantly modify your question it is however known.

More details:

I believe that it would be widely believed that all $S_{n,k}$ contain infinitely many prime numbers. Yet, that this is unproved.

The classical conjecture that there are infinitely many twin primes is the assertion that this set is infinite for $n=k=1$, the case that you exclude (possibly for this reason). It is also classically conjectured (Polignac's conjecture (1849)) that there are infinitely many primes with difference $2k$ for any fixed $k$, which is equivalent to the infinitude of this set for $n=1$.

Now, asymptotically there are the more numbers the product of exactly $n$ prime numbers the larger the $n$, specifically the counting function's order is $$ \frac{x}{\log x} (\log \log x)^{n-1} $$ So, it seems the more likely that $S_{n,k}$ contains infinitely many primes the larger the $n$.

Indeed, there is a famous result of Chen that for each even $h$ there exists infinitely many primes $p$ such that $p+h$ is a prime or the product of two primes. So if you would relax your condition to product of at most $n$ primes than indeed it would be known that the sets in question are infinite.

However, this exactly or at most is (as far as I understand) a severe problem.