Non-Wieferich primes under the abc conjecture

Ding, Yuchen

doi:10.1016/j.crma.2019.05.007

Number theory

Non-Wieferich primes under the abc conjecture
[La conjecture abc et les nombres premiers qui ne satisfont pas la condition de Wieferich]

Présenté par : the Editorial Board

Ding, Yuchen ¹

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

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

Résumé
Abstract

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.

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.

Reçu le : 2019-04-10
Accepté le : 2019-05-17
Publié le : 2019-06-01

DOI : 10.1016/j.crma.2019.05.007

Affiliations des auteurs :

Ding, Yuchen ¹

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

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     title = {Non-Wieferich primes under the abc conjecture},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {483--486},
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Ding, Yuchen. Non-Wieferich primes under the abc conjecture. Comptes Rendus. Mathématique, Tome 357 (2019) no. 6, pp. 483-486. doi : 10.1016/j.crma.2019.05.007. http://www.numdam.org/articles/10.1016/j.crma.2019.05.007/

Bibliographie
Cité par

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

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

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

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

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