Unexpected biases in the distribution of consecutive primes

Robert J. Lemke Oliver, Kannan Soundararajan

Published 2016-03-11Version 1

While the sequence of primes is very well distributed in the reduced residue classes $\pmod q$, the distribution of pairs of consecutive primes among the permissible $\phi(q)^2$ pairs of reduced residue classes $\pmod q$ is surprisingly erratic. This paper proposes a conjectural explanation for this phenomenon, based on the Hardy-Littlewood conjectures. The conjectures are then compared to numerical data, and the observed fit is very good.