Nonuniform Distributions of Residues of Prime Sequences in Prime Moduli

TL;DR

This paper investigates the distribution patterns of consecutive primes modulo q, revealing nonuniformities and connecting empirical observations with conjectures like Hardy-Littlewood.

Contribution

It introduces a new perspective on prime residue patterns, linking numerical evidence to conjectural asymptotics and extending understanding of prime distributions.

Findings

Patterns of prime residues are nonuniform and show preferences.

Computed frequencies of certain prime patterns are significantly less than expected.

Proposes a connection between empirical data and the Hardy-Littlewood prime k-tuple conjecture.

Abstract

For positive integers $q$, Dirichlet's theorem states that there are infinitely many primes in each reduced residue class modulo $q$. A stronger form of the theorem states that the primes are equidistributed among the $\varphi(q)$ reduced residue classes modulo $q$. This paper considers patterns of sequences of consecutive primes $(p_n, p_{n+1}, \ldots, p_{n+k})$ modulo $q$. Numerical evidence suggests a preference for certain prime patterns. For example, computed frequencies of the pattern $(a,a)$ modulo $q$ up to $x$ are much less than the expected frequency $\pi(x)/\varphi(q)^2$. We begin to rigorously connect the Hardy-Littlewood prime $k$-tuple conjecture to a conjectured asymptotic formula for the frequencies of prime patterns modulo $q$.