For which sets of residue classes are there easy elementary proofs that there are infinitely many primes in them, which don’t require the machinery of proofs of Dirichlet’s theorem?
Example: it’s simple to prove there are infinitely many primes ending in a digit that is either 3 or 7, because if there were finitely many you could multiply them all together, add a small constant, consider the prime factors of that, and get a contradiction. But what about if you just wanted to show there were infinitely many primes ending in 3, not “3 or 7”? When is that kind of thing relatively easy?
