Unexpected biases in the distribution of consecutive primes

Abstract

While the sequence of primes is very well distributed in the reduced residue classes (mod $q$), the distribution of pairs of consecutive primes among the permissible $\varphi (q{)}^2$ pairs of reduced residue classes (mod $q$) 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.

v4: Added dedication v3: Minor revisions to exposition, fixed one typo in Section 5 v2: Added references

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