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by Kevin Ford, Ben Green, Sergei Konyagin, James Maynard and Terence Tao;
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J. Amer. Math. Soc. 31 (2018), 65-105
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DOI: https://doi.org/10.1090/jams/876
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Published electronically: February 23, 2017
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Abstract: Let $p_n$ denote the $n$th prime. We prove that \[ \max _{p_{n} \leqslant X} (p_{n+1}-p_n) \gg \frac {\log X \log \log X\log \log \log \log X}{\log \log \log X}\] 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. References
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Bibliographic Information
- Kevin Ford
- Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61801
- MR Author ID: 325647
- ORCID: 0000-0001-9650-725X
- 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
- MR Author ID: 188475
- Email: konyagin@mi.ras.ru
- James Maynard
- Affiliation: Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom
- MR Author ID: 1007204
- Email: james.alexander.maynard@gmail.com
- Terence Tao
- Affiliation: Department of Mathematics, UCLA, 405 Hilgard Avenue, Los Angeles, California 90095
- MR Author ID: 361755
- ORCID: 0000-0002-0140-7641
- Email: tao@math.ucla.edu
- 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.
- © Copyright 2017 American Mathematical Society
- 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
- MathSciNet review: 3718451
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