Nonuniform Distributions of Patterns of Sequences of Primes in Prime Moduli

For positive integers q, Dirichlet’s theorem states that there are infinitely many primes in each reduced residue class modulo q. Extending a proof of Dirichlet’s theorem shows that the primes are equidistributed among the φ(q) reduced residue classes modulo q. This project considers patterns of sequences of consecutive primes (pn, pn+1, . . . , pn+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. We extend a data gathering procedure to estimate prime patterns up to 1018, an improvement of 8 orders of magnitude over previous methods. Using the extended range of data, a possible lower order term in the conjectured formula is identified via curve fitting. We begin to extend a numerical model to reduce the uncertainty in the predictions of these biases in prime patterns. The improved numerical could guide future progress towards understanding implications of the Hardy-Littlewood prime k-tuple conjecture.