Comptes Rendus
no. 6
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

Yuchen Ding1
1 Department of Mathematics, Nanjing University, Nanjing 210093, People's Republic of China
Comptes Rendus. Mathématique, Volume 357 (2019) no. 6, pp. 483-486.

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 ap−1≢1(modp2). 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 ap−1≢1(modp2) et p≡1(modk). 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 ap−1≢1(modp2). In this paper, we show that, for any given integers a⩾2 and k⩾2, there still are ≫log⁡x primes p⩽x satisfying ap−1≢1(modp2) and p≡1(modk), under the assumption of the abc conjecture. This improves a recent result of Chen and Ding.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2019.05.007

Yuchen Ding 1

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

[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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