Nonuniform Distributions of Residues of Prime Sequences in Prime Moduli

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 $q$.