Some exponential diophantine equations II: The equationx2–Dy2=kzfor evenk-Reference-Cited by-同舟云学术

Some exponential diophantine equations II: The equationx²–Dy²=k^zfor evenk

Published:2022-03-28 Issue:2 Volume:72 Page:341-354
ISSN:0139-9918
Container-title:Mathematica Slovaca
language:en
Short-container-title:

Author:

Fujita Yasutsugu¹,Le Maohua²

Affiliation:

1. Department of Mathematics , College of Industrial Technology, Nihon University , 2-11-1 Shin-ei, Narashino , Chiba , Japan

2. Institute of Mathematics , Lingnan Normal College , Zhangjiang , Guangdong, 524048 , China

Abstract

AbstractLetDbe a nonsquare integer, and letkbe an integer with |k| ≥ 1 and gcd(D,k) = 1. In the partIof this paper, using some properties on the representation of integers by binary quadratic primitive forms with discriminant 4D, M.-H. Le gave a series of explicit formulas for the integer solutions (x,y,z) of the equationx2–Dy2=kz, gcd(x,y) = 1,z> 0 for 2 ∤kor |k| is a power of 2. In this part, we give similar results for the other cases ofk.

Publisher

Walter de Gruyter GmbH

Subject

General Mathematics

Link

https://www.degruyter.com/document/doi/10.1515/ms-2022-0023/pdf

Reference20 articles.

1. Abu Muriefah, F. S.—Bugeaud, Y.: The Diophantine equation x2 + C = yn: a brief overview, Rev. Colomb. Math. 40 (2006), 31–37.

2. Bérczes, A.—Pink, I.: On generalized Lebesgue-Ramanujan-Nagell equations, An. St. Univ. Ovidius Constanta 22 (2014), 51–71.

3. Bugeaud, Y.: On the Diophantine equation x2 – pm = ±yn, Acta Arith. 80 (1997), 213–223.

4. Bugeaud, Y.—Shorey, T. N.: On the number of solutions of the generalized Ramanujan-Nagell equation, J. Reine Angew. Math. 539 (2001), 55–74.

5. Gou, S.—Wang, T.: The Diophantine equation x2 + 2a17b = yn, Czechoslovak Math. J. 62 (2012), 645–654.

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