Analysis of Primes in Arithmetical Progressions $7n + k$ up to a Trillion

Abstract

Each prime number fits in one of six unique forms $7n + k$. Each form $7n + k$, for $1 \leq k \leq 6$ forms an arithmetical progression; each containing, as assured by Dirichlet's Theorem, infinite number of primes. This work analyzes occurrences of primes in all of these arithmetical progressions from different angles, both 10 power blockwise as well as absolutely in ranges going as high as $1,000,000,000,000$.