Frequencies of Successive Pairs of Prime Residues

Abstract

We consider statistical properties of the sequence of ordered pairs obtained by taking the sequence of prime numbers and reducing modulo $m$. Using an inclusion/exclusion argument and a cutoff of an infinite product suggested by Pólya, we obtain a heuristic formula for the "probability" that a pair of consecutive prime numbers of size approximately $x$ will be congruent to $(a, a + d)$ modulo $m$. We demonstrate some symmetries of our formula. We test our formula and some of its consequences against data for $x$ in various ranges.