# 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

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