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

Mahadi Ddamulira, Florian Luca

Research output: Contribution to journalArticleResearchpeer-review

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 $k=3$.
Original languageEnglish
Pages (from-to)156-195
Number of pages40
JournalJournal of Number Theory
Volume207
DOIs
Publication statusPublished - 27 Aug 2019

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Generalized Fibonacci numbers
Pell's equation
Integer
Term
Fibonacci sequence
Square free
p.m.
Exception

Keywords

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

ASJC Scopus subject areas

  • Algebra and Number Theory

Fields of Expertise

  • Information, Communication & Computing

Cite this

On the $x$--coordinates of Pell equations which are $k$--generalized Fibonacci numbers. / Ddamulira, Mahadi; Luca, Florian.

In: Journal of Number Theory, Vol. 207, 27.08.2019, p. 156-195.

Research output: Contribution to journalArticleResearchpeer-review

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