# 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 language English 156-195 40 Journal of Number Theory 207 27 Aug 2019 https://doi.org/10.1016/j.jnt.2019.07.006 E-pub ahead of print - 27 Aug 2019

### Fingerprint

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

In: Journal of Number Theory, Vol. 207, 01.02.2020, p. 156-195.

Research output: Contribution to journalArticleResearchpeer-review

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AU - Luca, Florian

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