# On the x-coordinates of Pell equations which are products of two Lucas numbers

Research output: Contribution to journalArticleResearchpeer-review

### Abstract

Let $\{L_n\}_{n\ge 0}$ be the sequence of Lucas numbers given by $L_0=2, ~ L_1=1$ and $L_{n+2}=L_{n+1}+L_n$ for all $n\ge 0$. 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 product of two Lucas numbers, with a few exceptions that we completely characterize.
Original language English 1-20 20 The Fibonacci Quarterly Accepted/In press - 2 Aug 2019

Pell's equation
Lucas numbers
Integer
Square free
p.m.
Exception

### Keywords

• Lucas number
• Pell equation
• Linear forms in logarithms
• Baker's method

### ASJC Scopus subject areas

• Algebra and Number Theory

### Fields of Expertise

• Information, Communication & Computing

### Cite this

In: The Fibonacci Quarterly, 02.08.2019, p. 1-20.

Research output: Contribution to journalArticleResearchpeer-review

@article{2c34766c3ac1490486f17d5957cb13e1,
title = "On the x-coordinates of Pell equations which are products of two Lucas numbers",
abstract = "Let $\{L_n\}_{n\ge 0}$ be the sequence of Lucas numbers given by $L_0=2, ~ L_1=1$ and $L_{n+2}=L_{n+1}+L_n$ for all $n\ge 0$. 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 product of two Lucas numbers, with a few exceptions that we completely characterize.",
keywords = "Lucas number, Pell equation, Linear forms in logarithms, Baker's method",
author = "Mahadi Ddamulira",
year = "2019",
month = "8",
day = "2",
language = "English",
pages = "1--20",
journal = "The Fibonacci Quarterly",
issn = "0015-0517",
publisher = "Fibonacci Association",

}

TY - JOUR

T1 - On the x-coordinates of Pell equations which are products of two Lucas numbers

AU - Ddamulira, Mahadi

PY - 2019/8/2

Y1 - 2019/8/2

N2 - Let $\{L_n\}_{n\ge 0}$ be the sequence of Lucas numbers given by $L_0=2, ~ L_1=1$ and $L_{n+2}=L_{n+1}+L_n$ for all $n\ge 0$. 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 product of two Lucas numbers, with a few exceptions that we completely characterize.

AB - Let $\{L_n\}_{n\ge 0}$ be the sequence of Lucas numbers given by $L_0=2, ~ L_1=1$ and $L_{n+2}=L_{n+1}+L_n$ for all $n\ge 0$. 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 product of two Lucas numbers, with a few exceptions that we completely characterize.

KW - Lucas number

KW - Pell equation

KW - Linear forms in logarithms

KW - Baker's method

UR - https://arxiv.org/abs/1906.06330

M3 - Article

SP - 1

EP - 20

JO - The Fibonacci Quarterly

JF - The Fibonacci Quarterly

SN - 0015-0517

ER -