Non-Wieferich primes under the abc conjecture

Yuchen Ding

doi:10.1016/j.crma.2019.05.007

Number theory

Non-Wieferich primes under the abc conjecture

Presented by: the Editorial Board

Yuchen Ding ¹

¹ Department of Mathematics, Nanjing University, Nanjing 210093, People's Republic of China

Comptes Rendus. Mathématique, Volume 357 (2019) no. 6, pp. 483-486.

Abstract
Résumé

Assuming the abc conjecture, Silverman proved that, for any given positive integer $a ⩾ 2$ , there are $≫ \log x$ primes $p ⩽ x$ such that $a^{p - 1} ≢ 1 (\mod p^{2})$ . In this paper, we show that, for any given integers $a ⩾ 2$ and $k ⩾ 2$ , there still are $≫ \log x$ primes $p ⩽ x$ satisfying $a^{p - 1} ≢ 1 (\mod p^{2})$ and $p \equiv 1 (\mod k)$ , under the assumption of the abc conjecture. This improves a recent result of Chen and Ding.

Admettant la conjecture abc, Silverman a montré que, pour tout entier $a ⩾ 2$ , il existe au moins $≫ \log x$ nombres premiers $p ⩽ x$ tels que $a^{p - 1} ≢ 1 (\mod p^{2})$ . Admettant toujours la conjecture abc, nous montrons ici que, pour tous entiers $a ⩾ 2$ et $k ⩾ 2$ donnés, il y a encore au moins $≫ \log x$ nombres premiers $p ⩽ x$ tels que $a^{p - 1} ≢ 1 (\mod p^{2})$ et $p \equiv 1 (\mod k)$ . Ceci améliore un résultat récent de Chen et Ding.

Received: 2019-04-10
Accepted: 2019-05-17
Published online: 2019-06-01

DOI: 10.1016/j.crma.2019.05.007

Author's affiliations:

Yuchen Ding ¹

¹ Department of Mathematics, Nanjing University, Nanjing 210093, People's Republic of China

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     author = {Yuchen Ding},
     title = {Non-Wieferich primes under the abc conjecture},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {483--486},
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     year = {2019},
     doi = {10.1016/j.crma.2019.05.007},
     language = {en},
}

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Yuchen Ding. Non-Wieferich primes under the abc conjecture. Comptes Rendus. Mathématique, Volume 357 (2019) no. 6, pp. 483-486. doi : 10.1016/j.crma.2019.05.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.05.007/

References
Cited by

[1] Y.-G. Chen; Y. Ding Non-Wieferich primes in arithmetic progressions, Proc. Amer. Math. Soc., Volume 145 (2017), pp. 1833-1836

[2] H. Graves; M.R. Murty The abc conjecture and non-Wieferich primes in arithmetic progressions, J. Number Theory, Volume 133 (2013), pp. 1809-1813

[3] J.H. Silverman Wieferich's criterion and the abc-conjecture, J. Number Theory, Volume 30 (1988), pp. 226-237

[4] A. Wieferich Zum letzten Fermatschen Theorem, J. Reine Angew. Math., Volume 136 (1909), pp. 293-302 (in German)

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