Geodesic
Non-Wieferich primes in number fields and $abc$-conjecture
Czechoslovak Mathematical Journal, Tome 68 (2018) no. 2, pp. 445-453
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

Let $K/\mathbb {Q}$ be an algebraic number field of class number one and let $\mathcal {O}_K$ be its ring of integers. We show that there are infinitely many non-Wieferich primes with respect to certain units in $\mathcal {O}_K$ under the assumption of the $abc$-conjecture for number fields.
Let $K/\mathbb {Q}$ be an algebraic number field of class number one and let $\mathcal {O}_K$ be its ring of integers. We show that there are infinitely many non-Wieferich primes with respect to certain units in $\mathcal {O}_K$ under the assumption of the $abc$-conjecture for number fields.
DOI : 10.21136/CMJ.2018.0485-16
Classification : 11A41, 11R04
Keywords: Wieferich prime; non-Wieferich prime; number field; $abc$-conjecture 
@article{10_21136_CMJ_2018_0485_16,
     author = {Kotyada, Srinivas and Muthukrishnan, Subramani},
     title = {Non-Wieferich primes in number fields and $abc$-conjecture},
     journal = {Czechoslovak Mathematical Journal},
     pages = {445--453},
     year = {2018},
     volume = {68},
     number = {2},
     doi = {10.21136/CMJ.2018.0485-16},
     mrnumber = {3819183},
     zbl = {06890382},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2018.0485-16/}
}
TY  - JOUR
AU  - Kotyada, Srinivas
AU  - Muthukrishnan, Subramani
TI  - Non-Wieferich primes in number fields and $abc$-conjecture
JO  - Czechoslovak Mathematical Journal
PY  - 2018
SP  - 445
EP  - 453
VL  - 68
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2018.0485-16/
DO  - 10.21136/CMJ.2018.0485-16
LA  - en
ID  - 10_21136_CMJ_2018_0485_16
ER  - 
%0 Journal Article
%A Kotyada, Srinivas
%A Muthukrishnan, Subramani
%T Non-Wieferich primes in number fields and $abc$-conjecture
%J Czechoslovak Mathematical Journal
%D 2018
%P 445-453
%V 68
%N 2
%U http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2018.0485-16/
%R 10.21136/CMJ.2018.0485-16
%G en
%F 10_21136_CMJ_2018_0485_16
Kotyada, Srinivas; Muthukrishnan, Subramani. Non-Wieferich primes in number fields and $abc$-conjecture. Czechoslovak Mathematical Journal, Tome 68 (2018) no. 2, pp. 445-453. doi: 10.21136/CMJ.2018.0485-16

[1] Graves, H., Murty, M. R.: The $abc$ conjecture and non-Wieferich primes in arithmetic progressions. J. Number Theory 133 (2013), 1809-1813. | DOI | MR | JFM

[2] Győry, K.: On the $abc$ conjecture in algebraic number fields. Acta Arith. 133 (2008), 281-295. | DOI | MR | JFM

[3] Murty, M. R.: The $ABC$ conjecture and exponents of class groups of quadratic fields. Number Theory. Proc. Int. Conf. On Discrete Mathematics and Number Theory, Tiruchirapalli, India, 1996 V. K. Murty et al. Contemp. Math. 210. AMS, Providence (1998), 85-95. | DOI | MR | JFM

[4] Murty, M. R., Esmonde, J.: Problems in Algebraic Number Theory. Graduate Texts in Mathematics 190, Springer, Berlin (2005). | DOI | MR | JFM

[5] PrimeGrid Project. Available at http://www.primegrid.com/</b>

[6] Silverman, J. H.: Wieferich's criterion and the $abc$-conjecture. J. Number Theory 30 (1988), 226-237. | DOI | MR | JFM

[7] Vojta, P.: Diophantine Approximations and Value Distribution Theory. Lecture Notes in Mathematics 1239, Springer, Berlin (1987). | DOI | MR | JFM

[8] Wieferich, A.: Zum letzten Fermatschen Theorem. J. Reine Angew. Math. 136 (1909), 293-302 German \99999JFM99999 40.0256.03. | DOI | MR

Cité par Sources :