Number theory
Non-Wieferich primes under the abc conjecture
[La conjecture abc et les nombres premiers qui ne satisfont pas la condition de Wieferich]
Comptes Rendus. Mathématique, Tome 357 (2019) no. 6, pp. 483-486.

Admettant la conjecture abc, Silverman a montré que, pour tout entier a2, il existe au moins logx nombres premiers px tels que ap11(modp2). Admettant toujours la conjecture abc, nous montrons ici que, pour tous entiers a2 et k2 donnés, il y a encore au moins logx nombres premiers px tels que ap11(modp2) et p1(modk). Ceci améliore un résultat récent de Chen et Ding.

Assuming the abc conjecture, Silverman proved that, for any given positive integer a2, there are logx primes px such that ap11(modp2). In this paper, we show that, for any given integers a2 and k2, there still are logx primes px satisfying ap11(modp2) and p1(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
Ding, Yuchen 1

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

[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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