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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Non-Wieferich primes in arithmetic progressions
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by Yong-Gao Chen and Yu Ding PDF
Proc. Amer. Math. Soc. 145 (2017), 1833-1836 Request permission

Abstract:

Graves and Murty proved that for any integer $a\ge 2$ and any fixed integer $k\ge 2$, there are $\gg \log x/\log \log x$ primes $p\le x$ such that $a^{p-1}\not \equiv 1\pmod {p^2}$ and $p\equiv 1\pmod k$, under the assumption of the abc conjecture. In this paper, for any fixed $M$, the bound $\log x/\log \log x$ is improved to $(\log x/\log \log x) (\log \log \log x)^M$.
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Additional Information
  • Yong-Gao Chen
  • Affiliation: School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, People’s Republic of China
  • MR Author ID: 304097
  • Email: ygchen@njnu.edu.cn
  • Yu Ding
  • Affiliation: School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, People’s Repubilc of China
  • MR Author ID: 1199999
  • Email: 840172236@qq.com
  • Received by editor(s): November 4, 2015
  • Received by editor(s) in revised form: February 25, 2016
  • Published electronically: January 23, 2017
  • Additional Notes: This work was supported by the National Natural Science Foundation of China (No. 11371195) and PAPD
  • Communicated by: Matthew A. Papanikolas
  • © Copyright 2017 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 1833-1836
  • MSC (2010): Primary 11A41, 11B25
  • DOI: https://doi.org/10.1090/proc/13201
  • MathSciNet review: 3611299