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 languageEnglish
Pages (from-to)1-20
Number of pages20
JournalThe Fibonacci Quarterly
Publication statusAccepted/In press - 2 Aug 2019

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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

On the $x$--coordinates of Pell equations which are products of two Lucas numbers. / Ddamulira, Mahadi.

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

Research output: Contribution to journalArticleResearchpeer-review

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