1. How much is known about primes studied by their index in the 1 mod 6 and 5 mod 6 sets?

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4. The question is typically asked in the form, "How many primes less than X are congruent to 1 mod 6?" The answer is asymptotically π(X)/2, where π(X) is the prime counting function. Indeed, the general answer to, "If gcd(a,q) = 1, how many primes less than X are congruent to a mod q?" is asymptotically π(X)/ø(q), where ø is supposed to be the Euler totient function. The fact that there are infinitely many primes in any such congruence class is known as Dirichlet's Theorem on Arithmetic Progressions, and the precise asymptotic statement is called the Prime Number Theorem for Arithmetic Progressions.

To answer your question, if you listed out the positive integers congruent to a mod q, you'd find that the n-th prime in the sequence would have index about n log n.

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