Unexpected biases in the distribution of consecutive primes.

ABSTRACT

Although the sequence of primes is very well distributed in the reduced residue classes [Formula see text], the distribution of pairs of consecutive primes among the permissible ϕ(q)(2) pairs of reduced residue classes [Formula see text] is surprisingly erratic. This paper proposes a conjectural explanation for this phenomenon, based on the Hardy-Littlewood conjectures. The conjectures are then compared with numerical data, and the observed fit is very good.