A Conjecture on Primes in Arithmetic Progressions and Geometric Intervals

Dirichlet’s theorem on primes in arithmetic progressions states that for any positive integer q and any coprime integer a, there are infinitely many primes in the arithmetic progression a + nq (n ∈ N). Unfortunately, the theorem does not predict the gap between those primes. Several renowned open questions such as the size of Linnik’s constant try to say more about the distribution of such primes. This manuscript postulates a weak, but explicit, generalization of Linnik’s theorem, namely that each geometric interval [q^t, q^(t+1)] contains a prime from each coprime congruence class modulo q ∈ N≥2. Subsequently, it proves the conjecture theoretically for all sufficiently large t, as well as computationally for all sufficiently small q. Finally, the impact of this conjecture on a result of Pomerance related to Carmichael’s conjecture is outlined.