Lucas sequences whose 12th or 9th term is a square

Andrew Bremner, N. Tzanakis

Research output: Contribution to journalArticle

11 Scopus citations

Abstract

Let P and Q be non-zero relatively prime integers. The Lucas sequence {Un(P,Q)} is defined by U0=0, U1=1, Un=PUn-1-QUn-2 (n≥2). We show that the only sequence with U12(P,Q) a perfect square is the Fibonacci sequence {Un(1,-1)}; and we show that there are no non-degenerate sequences {Un(P,Q)} with U9(P,Q) a perfect square. The argument involves finding all rational points on several curves of genus 2.

Original languageEnglish (US)
Pages (from-to)215-227
Number of pages13
JournalJournal of Number Theory
Volume107
Issue number2
DOIs
StatePublished - Aug 1 2004

Keywords

  • Elliptic curve
  • Formal group
  • Lucas sequence

ASJC Scopus subject areas

  • Algebra and Number Theory

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