On the x-coordinates of Pell equations which are k-generalized Fibonacci numbers

JOURNAL OF NUMBER THEORY

卷:207

Article

Ddamulira, Mahadi¹  Luca, Florian^2,3,4,5 

[1] Graz Univ Technol, Inst Anal & Number Theory, Kopernikusgasse 24-2, A-8010 Graz, Austria

[2] Univ Witwatersrand, Sch Math, Private Bag X3, ZA-2050 Johannesburg, South Africa

[3] Res Grp Algebra Struct & Applicat, Jeddah, Saudi Arabia

[4] Univ Ostrava, Fac Sci, Dept Math, 30 Dubna 22, CZ-70103 Ostrava 1, Czech Republic

[5] UNAM, Ctr Ciencias Matemat, Morelia, Michoacan, Mexico

关键词: Pell equation;    Generalized Fibonacci sequence;    Linear form in logarithms;    Reduction method;   

DOI  :  10.1016/j.jnt.2019.07.006

来源: Elsevier

【 摘 要 】

For an integer k >= 2, let {F-n(k)}(n >= 2-k) be the k-generalized Fibonacci sequence which starts with 0, ..., 0, 1 (a total of k terms) and for which each term afterwards is the sum of the k preceding terms. In this paper, for an integer d >= 2 which is square-free, we show that there is at most one value of the positive integer x participating in the Pell equation x(2) - dy(2) = +/- 1, which is a k-generalized Fibonacci number, with a couple of parametric exceptions which we completely characterize. This paper extends previous work from [18] for the case k = 2 and [17] for the case k = 3. (C) 2019 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

【 授权许可】

Free