For any $a \in \Bbb{Z}$, can we always find two prime numbers $p, q$, such that $p - q \in (a)$?

Question

This is a major weakening of many prime sum / difference existence questions.

Let $a \in \Bbb{Z}$ and $(a)$ the ideal generated by $a$. Then do there exist two primes $p, q$ such that $p - q \in (a)$ at least?

Thanks.

Answer 1

This can be done in an elementary way; Dirichlet is extreme overkill.

To see this, note that there are infinitely many primes, but only finitely many remainders on division by $a$. By the pigeonhole principle, there are two primes $p$ and $q$ with the same remainder, and we are done.