I am trying to prove there is no infinite arithmetic progression involving only prime numbers. (In other words, I want to prove that if $a, b \in \mathbb{N}$, then there exists some $n$ such that $a + bn$ is not prime).
My approach: I think that taking a prime $p$ that is coprime to $b$, and showing that it divides many terms in the sequence is a viable way to go about proving this, but I can't see a clear way to write/implement this idea. Any advice?
This post is very similar, but I don't understand the answer proof very well. I would prefer to use the method I described above, if possible.
Best Answer
There are arbitrarily long sequences of consecutive composites. The well known proof: look at $$ k!+2, k!+3, \ldots, k! + k . $$
So no arithmetic progression can contain only primes.