Solutions of polynomial Pell's equation

H. Yokota

Research output: Contribution to journalArticle

1 Citation (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

Keywords

  • Polynomial Pell's equation

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

Solutions of polynomial Pell's equation. / Yokota, H.

In: Journal of Number Theory, Vol. 130, No. 9, 09.2010, p. 2003-2010.

Research output: Contribution to journalArticle

Yokota, H. / Solutions of polynomial Pell's equation. In: Journal of Number Theory. 2010 ; Vol. 130, No. 9. pp. 2003-2010.
@article{dbf4393ea75a40ab9286f52854e5df12,
title = "Solutions of polynomial Pell's equation",
abstract = "Let D=F2+2G be a monic quartic polynomial in Z[x], where degG",
keywords = "Polynomial Pell's equation",
author = "H. Yokota",
year = "2010",
month = "9",
doi = "10.1016/j.jnt.2009.12.006",
language = "English",
volume = "130",
pages = "2003--2010",
journal = "Journal of Number Theory",
issn = "0022-314X",
publisher = "Academic Press Inc.",
number = "9",

}

TY - JOUR

T1 - Solutions of polynomial Pell's equation

AU - Yokota, H.

PY - 2010/9

Y1 - 2010/9

N2 - Let D=F2+2G be a monic quartic polynomial in Z[x], where degG

AB - Let D=F2+2G be a monic quartic polynomial in Z[x], where degG

KW - Polynomial Pell's equation

UR - http://www.scopus.com/inward/record.url?scp=77953684918&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77953684918&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2009.12.006

DO - 10.1016/j.jnt.2009.12.006

M3 - Article

VL - 130

SP - 2003

EP - 2010

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 9

ER -