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

Mahadi Ddamulira*, Florian Luca

*Korrespondierende/r Autor/in für diese Arbeit

Publikation: Beitrag in einer FachzeitschriftArtikel

Abstract

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.

Originalspracheenglisch
Seiten (von - bis)156-195
Seitenumfang40
FachzeitschriftJournal of Number Theory
Jahrgang207
Frühes Online-Datum27 Aug 2019
DOIs
PublikationsstatusVeröffentlicht - 1 Feb 2020

ASJC Scopus subject areas

  • !!Algebra and Number Theory

Fields of Expertise

  • Information, Communication & Computing

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