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

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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)18-37
JournalThe Fibonacci Quarterly
Volume58
Issue number1
DOIs
Publication statusPublished - 13 Feb 2020

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

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