Nonuniform Distributions of Residues of Prime Sequences in Prime Moduli

2019/08/19 by David Wu, Wu, David
Computer Science · Mathematics · #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1908.07095

openalex publication_date 2019/08/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 φ(q) reduced residue classes modulo q. This paper considers patterns of sequences of consecutive primes (p_n, p_n+1, …, 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 π(x)/φ(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.

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