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Non-Wieferich primes in arithmetic progressions


Authors: Yong-Gao Chen and Yu Ding
Journal: Proc. Amer. Math. Soc. 145 (2017), 1833-1836
MSC (2010): Primary 11A41, 11B25
DOI: https://doi.org/10.1090/proc/13201
Published electronically: January 23, 2017
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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
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
Email: 840172236@qq.com

DOI: https://doi.org/10.1090/proc/13201
Keywords: Wieferich primes, arithmetic progressions, abc conjecture
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
Article copyright: © Copyright 2017 American Mathematical Society