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Unexpected biases in the distribution of consecutive primes

Number Theory 2022-04-27 v4

Abstract

While the sequence of primes is very well distributed in the reduced residue classes (mod qq), the distribution of pairs of consecutive primes among the permissible ϕ(q)2\phi(q)^2 pairs of reduced residue classes (mod qq) 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.

Keywords

Cite

@article{arxiv.1603.03720,
  title  = {Unexpected biases in the distribution of consecutive primes},
  author = {Robert J. Lemke Oliver and Kannan Soundararajan},
  journal= {arXiv preprint arXiv:1603.03720},
  year   = {2022}
}

Comments

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