question about the forms of prime numbers

Question

I was thinking about primes earlier and I thought of a hypothesis that I have been unable to prove. I was wondering whether it was a known theorem and whether anyone knows a proof or can prove (or disprove) it.

here it is:

there are an infinite number of primes that satisfy the equation:

$p \equiv a \bmod n$

for all A and N where A and N are relatively prime.

Answer 1

This is Dirichlet's theorem on arithmetic progressions of primes. A brief sketch of the proof is given in the article, along with a reference to Jürgen Neukirch's Algebraic number theory $(1999)$.