The x-coordinates of Pell equations and sums of two Fibonacci numbers II

Mahadi Ddamulira; Florian Luca

doi:10.1007/s12044-020-00578-4

The x-coordinates of Pell equations and sums of two Fibonacci numbers II

Mahadi Ddamulira^*, Florian Luca

^*Corresponding author for this work

Institute of Analysis and Number Theory (5010)

Research output: Contribution to journal › Article › peer-review

Abstract

Let {Fn}n≥0 be the sequence of Fibonacci numbers defined by F= 0 , F₁= 1 and F_n₊₂= F_n₊₁+ F_n for all n≥ 0. 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 x²- dy²= ± 4 which is a sum of two Fibonacci numbers, with a few exceptions that we completely characterize.

Original language	English
Article number	58
Number of pages	21
Journal	Proceedings of the Indian Academy of Sciences: Mathematical Sciences
Volume	130
Issue number	1
Early online date	23 Sept 2020
DOIs	https://doi.org/10.1007/s12044-020-00578-4
Publication status	Published - 1 Dec 2020

Keywords

Fibonacci number
linear form in logarithm
Pell equation
reduction method

ASJC Scopus subject areas

Mathematics(all)

Fields of Expertise

Information, Communication & Computing

Access to Document

10.1007/s12044-020-00578-4

https://www.ias.ac.in/article/fulltext/pmsc/130/0058Licence: CC BY-NC-ND 4.0

Cite this

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AB - Let {Fn}n≥0 be the sequence of Fibonacci numbers defined by F= 0 , F1= 1 and Fn+2= Fn+1+ Fn for all n≥ 0. 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 x2- dy2= ± 4 which is a sum of two Fibonacci numbers, with a few exceptions that we completely characterize.

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KW - linear form in logarithm

KW - Pell equation

KW - reduction method

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The x-coordinates of Pell equations and sums of two Fibonacci numbers II

Abstract

Keywords

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