| Title:
|
Non-Wieferich primes in number fields and $abc$-conjecture (English) |
| Author:
|
Kotyada, Srinivas |
| Author:
|
Muthukrishnan, Subramani |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
68 |
| Issue:
|
2 |
| Year:
|
2018 |
| Pages:
|
445-453 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Wieferich prime |
| Keyword:
|
non-Wieferich prime |
| Keyword:
|
number field |
| Keyword:
|
$abc$-conjecture |
| MSC:
|
11A41 |
| MSC:
|
11R04 |
| idZBL:
|
Zbl 06890382 |
| idMR:
|
MR3819183 |
| DOI:
|
10.21136/CMJ.2018.0485-16 |
| . |
| Date available:
|
2018-06-11T10:54:44Z |
| Last updated:
|
2020-07-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147228 |
| . |
| Reference:
|
[1] Graves, H., Murty, M. R.: The $abc$ conjecture and non-Wieferich primes in arithmetic progressions.J. Number Theory 133 (2013), 1809-1813. Zbl 1272.11014, MR 3027939, 10.1016/j.jnt.2012.10.012 |
| Reference:
|
[2] Győry, K.: On the $abc$ conjecture in algebraic number fields.Acta Arith. 133 (2008), 281-295. Zbl 1188.11011, MR 2434605, 10.4064/aa133-3-6 |
| Reference:
|
[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. Zbl 0893.11043, MR 1478486, 10.1090/conm/210/02785 |
| Reference:
|
[4] Murty, M. R., Esmonde, J.: Problems in Algebraic Number Theory.Graduate Texts in Mathematics 190, Springer, Berlin (2005). Zbl 1055.11001, MR 2090972, 10.1007/b138452 |
| Reference:
|
[5] : PrimeGrid Project. Available at http://www.primegrid.com/.. |
| Reference:
|
[6] Silverman, J. H.: Wieferich's criterion and the $abc$-conjecture.J. Number Theory 30 (1988), 226-237. Zbl 0654.10019, MR 0961918, 10.1016/0022-314X(88)90019-4 |
| Reference:
|
[7] Vojta, P.: Diophantine Approximations and Value Distribution Theory.Lecture Notes in Mathematics 1239, Springer, Berlin (1987). Zbl 0609.14011, MR 0883451, 10.1007/BFb0072989 |
| Reference:
|
[8] Wieferich, A.: Zum letzten Fermatschen Theorem.J. Reine Angew. Math. 136 (1909), 293-302 German \99999JFM99999 40.0256.03. MR 1580782, 10.1515/crll.1909.136.293 |
| . |
