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Chains of Large Gaps Between Primes

  • Kevin FordEmail author
  • James Maynard
  • Terence Tao
Chapter

Abstract

Let pn denote the n-th prime, and for any \(k \geqslant 1\) and sufficiently large X, define the quantity
$$\displaystyle G_k(X) := \max _{p_{n+k} \leqslant X} \min ( p_{n+1}-p_n, \dots , p_{n+k}-p_{n+k-1} ), $$
which measures the occurrence of chains of k consecutive large gaps of primes. Recently, with Green and Konyagin, the authors showed that
$$\displaystyle G_1(X) \gg \frac {\log X \log \log X\log \log \log \log X}{\log \log \log X} $$
for sufficiently large X. In this note, we combine the arguments in that paper with the Maier matrix method to show that
$$\displaystyle G_k(X) \gg \frac {1}{k^2} \frac {\log X \log \log X\log \log \log \log X}{\log \log \log X} $$
for any fixed k and sufficiently large X. The implied constant is effective and independent of k.

Notes

Acknowledgements

Kevin Ford thanks the hospitality of the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences. The research of James Maynard was conducted partly while he was a CRM-ISM postdoctoral fellow at the Université de Montréal, and partly while he was a Fellow by Examination at Magdalen College, Oxford.

Kevin Ford was supported by NSF grants DMS-1201442 and DMS-1501982. Terence Tao was supported by a Simons Investigator grant, the James and Carol Collins Chair, the Mathematical Analysis & Application Research Fund Endowment, and by NSF grant DMS-1266164.

The authors thank Tristan Freiberg for some corrections.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Mathematical InstituteOxfordEngland
  3. 3.Department of MathematicsUCLALos AngelesUSA

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