Sequence containing infinite prime numbers

Question

Let’s us take an infinite set of positive integer numbers defined with certain relation $\{ a,b,...\}$. Iif we can prove it contains infinitely many numbers of the form $a+nd$, where $a,d$ are coprime, then will it be okay to say it contains infinitely many prime numbers as stated by Dirichlet's theoremm?

Answer 1

To see that this is false, start with the sequence $\{4k+1\}_{k=0}^{\infty}$, so $a=4, d=1$ in your notation. By Dirichlet, that contains infinitely many primes.

Of course, it also contains infinitely many non primes:

(Pf: say $p=4k+1$ is prime. Then $p\,|\,(4(k+np)+1)$ $\;\;\forall n$.)

Let $S$ be the sequence of all non-primes congruent to $1 \pmod 4$. $S$ is a counterexample to your desired claim.

A similar argument works for every arithmetic progression.