On squares in Lucas sequences

Andrew Bremner, N. Tzanakis

Research output: Contribution to journalArticle

10 Scopus citations

Abstract

Let P and Q be non-zero integers. The Lucas sequence {Un (P, Q)} is defined by U0 = 0, U1 = 1, Un = P Un - 1 - Q Un - 2 (n ≥ 2). The question of when Un (P, Q) can be a perfect square has generated interest in the literature. We show that for n = 2, ..., 7, Un is a square for infinitely many pairs (P, Q) with gcd (P, Q) = 1; further, for n = 8, ..., 12, the only non-degenerate sequences where gcd (P, Q) = 1 and Un (P, Q) = □, are given by U8 (1, - 4) = 212, U8 (4, - 17) = 6202, and U12 (1, - 1) = 122.

Original languageEnglish (US)
Pages (from-to)511-520
Number of pages10
JournalJournal of Number Theory
Volume124
Issue number2
DOIs
StatePublished - Jun 1 2007

Keywords

  • Genus two curves
  • Lucas sequence
  • Squares

ASJC Scopus subject areas

  • Algebra and Number Theory

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