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

Mahadi Ddamulira, Florian Luca

Publikation: Beitrag in einer FachzeitschriftArtikelForschungBegutachtung

Abstract

For an integer $k\geq 2$, let $\{F^{(k)}_{n}\}_{n\geqslant 2-k}$ be the $ k$--generalized Fibonacci sequence which starts with $0, \ldots, 0,1$ (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 $d\geq 2$ which is square free, we show that there is at most one value of the positive integer $x$ participating in the Pell equation $x^{2}-dy^{2} =\pm 1$ which is a $k$--generalized Fibonacci number, with a couple of parametric exceptions which we completely characterise. This paper extends previous work from [17] for the case $k=2$ and [16] 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

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    ASJC Scopus subject areas

    • !!Algebra and Number Theory

    Fields of Expertise

    • Information, Communication & Computing

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