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Long gaps between primes


Authors: Kevin Ford, Ben Green, Sergei Konyagin, James Maynard and Terence Tao
Journal: J. Amer. Math. Soc. 31 (2018), 65-105
MSC (2010): Primary 11N05; Secondary 11N36, 05C65, 05C70
DOI: https://doi.org/10.1090/jams/876
Published electronically: February 23, 2017
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Abstract: Let $ p_n$ denote the $ n$th prime. We prove that

$\displaystyle \max _{p_{n} \leq X} (p_{n+1}-p_n) \gg \frac {\log X \log \log X\log \log \log \log X}{\log \log \log X}$

for sufficiently large $ X$, improving upon recent bounds of the first, second, third, and fifth authors and of the fourth author. Our main new ingredient is a generalization of a hypergraph covering theorem of Pippenger and Spencer, proven using the Rödl nibble method.

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Additional Information

Kevin Ford
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61801
Email: ford@math.uiuc.edu

Ben Green
Affiliation: Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom
Email: ben.green@maths.ox.ac.uk

Sergei Konyagin
Affiliation: Steklov Mathematical Institute, 8 Gubkin Street, Moscow, 119991, Russia
Email: konyagin@mi.ras.ru

James Maynard
Affiliation: Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom
Email: james.alexander.maynard@gmail.com

Terence Tao
Affiliation: Department of Mathematics, UCLA, 405 Hilgard Avenue, Los Angeles, California 90095
Email: tao@math.ucla.edu

DOI: https://doi.org/10.1090/jams/876
Received by editor(s): December 23, 2015
Received by editor(s) in revised form: November 10, 2016, and December 21, 2016
Published electronically: February 23, 2017
Additional Notes: The first author was supported by NSF grants DMS-1201442 and DMS-1501982.
The second author was supported by ERC Starting Grant 279438, Approximate algebraic structure, and by a Simons Investigator grant.
The fifth author was supported by a Simons Investigator grant, by the James and Carol Collins Chair, by the Mathematical Analysis & Application Research Fund Endowment, and by NSF grant DMS-1266164.
Article copyright: © Copyright 2017 American Mathematical Society

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