Solutions of polynomial Pell's equation

H. Yokota

Research output: Contribution to journalReview article

2 Citations (Scopus)

Abstract

Let D=F2+2G be a monic quartic polynomial in Z[x], where degG<degF. Then for F/G∈Q[x], a necessary and sufficient condition for the solution of the polynomial Pell's equation X2-DY2=1 in Z[x] has been shown. Also, the polynomial Pell's equation X2-DY2=1 has nontrivial solutions X,Y∈Q[x] if and only if the values of period of the continued fraction of D are 2, 4, 6, 8, 10, 14, 18, and 22 has been shown. In this paper, for the period of the continued fraction of D is 4, we show that the polynomial Pell's equation has no nontrivial solutions X,Y∈Z[x].

Original languageEnglish
Pages (from-to)2003-2010
Number of pages8
JournalJournal of Number Theory
Volume130
Issue number9
DOIs
Publication statusPublished - 2010 Sep 1

Keywords

  • Polynomial Pell's equation

ASJC Scopus subject areas

  • Algebra and Number Theory

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