I'm faced with the following problem on primes. Does someone have any clue? Is it (a reformulation of) an open problem?


Let $d$ be a positive integer, $d\geq 2$. By Dirichlet's theorem, there is an infinite set $\mathcal{P}$ of primes congruent to $1$ modulo $d$. 


Consider the set $N$ of integers $n=1+kd$ where all prime divisors of $k$ belong to $\mathcal{P}$. 


Could we expect that $N$ contains infinitely many primes? (we need at least $d$ to be even).



Infinitely many primes in particular progressions
Yes, we should expect it.
For any even $d \ge 2$, Dickson's conjecture implies that there are infinitely many primes $p \equiv 1 \bmod d$ such that $1 + p d$ is prime.


Of course, expecting and proving are very different matters.


