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2006 Fiscal Year Final Research Report Summary

On solutions of polynomial Pell's equations and the continued fraction factorization algorithm

Research Project

Project/Area Number 17540052
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionShibaura Institute of Technology (2006)
Hiroshima Institute of Technology (2005)

Principal Investigator

YOKOTA Hisashi  Shibaura Inst of Tech, Engineering, Professor, 工学部, 教授 (90210616)

Project Period (FY) 2005 – 2006
KeywordsPolynomial Pell's equation / elliptic curve / continued fraction
Research Abstract

Study on the polynomial Pell's equation was first done by Abel in the connection of finding an elliptic integral which can be expressed using an elementary function. This was done in the field of rational numbers.
When we restrict solutions of the polynomial Pell's equation to be integer coefficient polynomial, the known result is only for a monic quartic polynomial. We have shown in 2003, a necessary and sufficient condition for the polynomial Pell's equation has a nontrivial integer coefficient polynomial solution for D = A^2+2C and A/C∈Q[x].
In this research, collaborating with Prof.Webb, we have studied the polynomial Pell's equation using the period of continued fraction expansions of √<D> in the connection with rational points on the elliptic curve arising from the partial quotients. We also have studied the polynomial Pell's equation by looking at the small periods.
For D a monic quartic polynomial, we are able to show that there is no period 3 continued fraction expansion.
For D a monic polynomial, we are able to show that the values of period of continued fraction expansions are even if and only if the polynomial Pell's equation X^2-DY^2 = 1 has a nontrivial solution.
For D a monic quartic polynomial, we are able to show that the polynomial Pell's equation X^2-DY^2 = 1 has a nontrivial solution in Q[x] if and only if the values of the period of continued fraction expansions are 2,4,6,8,10,14,18,22.

  • Research Products

    (4 results)

All 2007 2006 2005

All Journal Article (3 results) Patent(Industrial Property Rights) (1 results)

  • [Journal Article] Polynomial Pell' s equation and periods of quadratic irrationals2007

    • Author(s)
      H.Yokota
    • Journal Title

      JP Journal of Algebra, Number Theory and Applications 8

      Pages: 135-144

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Polynomial Pell's equation and periods of quadratic irrationals2007

    • Author(s)
      H.Yokota
    • Journal Title

      JP Journal of Algebra, Number Theory and Applications 8

      Pages: 135-144

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On the period of continued fraction2006

    • Author(s)
      W.Webb, H.Yokota
    • Journal Title

      JP Journal of Algebra, Number Theory and Applications 6

      Pages: 551-559

    • Description
      「研究成果報告書概要(和文)」より
  • [Patent(Industrial Property Rights)] 数学問題解答評価方法2005

    • Inventor(s)
      横田 壽
    • Industrial Property Rights Holder
      横田 壽
    • Industrial Property Number
      公開特許,P06P005221
    • Filing Date
      2005-08-02
    • Acquisition Date
      2007-02-15
    • Description
      「研究成果報告書概要(和文)」より

URL: 

Published: 2008-05-27  

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