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

Mahadi Ddamulira*, Florian Luca

*Corresponding author for this work

Research output: Contribution to journalArticle

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.

Original languageEnglish
Pages (from-to)156-195
Number of pages40
JournalJournal of Number Theory
Volume207
Early online date27 Aug 2019
DOIs
Publication statusPublished - 1 Feb 2020

Keywords

  • Pell equation
  • Generalized Fibonacci sequence
  • Linear forms in logarithms
  • Baker's method
  • Reduction method
  • Linear form in logarithms

ASJC Scopus subject areas

  • Algebra and Number Theory

Fields of Expertise

  • Information, Communication & Computing

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