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Application of continued fractions to real quadratic fields

Research Project

Project/Area Number 22540030
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionGakushuin University

Principal Investigator

KAWAMOTO Fuminori  学習院大学, 理学部, 助教 (50195161)

Co-Investigator(Kenkyū-buntansha) TOMITA Koshi  名城大学, 理工学部, 准教授 (50300207)
Co-Investigator(Renkei-kenkyūsha) KISHI Yasuhiro  愛知教育大学, 教育学部, 准教授 (60380375)
SUZUKI Hiroshi  名古屋大学, 大学院多元数理科学研究科, 准教授 (70235993)
Project Period (FY) 2010-04-01 – 2014-03-31
Project Status Completed (Fiscal Year 2013)
Budget Amount *help
¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000)
Fiscal Year 2013: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2012: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2011: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2010: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Keywordsガウス予想 / 類数 / 基本単数 / 連分数 / 極小型実二次体 / 極小型自然数 / 末尾急増型主要対称部分 / 増殖分解 / 連分数展開の周期 / 実二次体
Research Abstract

(1) Kawamoto-Tomita(2012) posed a conjecture by using numerical datas that the minimal element of each period gives a real quadratic field with class number 1 of minimal type. (2) We examine to construct such a field in each even period, and then Kawamoto-Kishi-Tomita(2014) gave a way of constructing positive integers with even period of minimal type. (3) By using this construction, we introduce notions of ``extremely large end (ELE)'' for a primary symmetric part and of ``pre-ELE type'' for a finite string. Then we give a way of constructing primary symmetric parts of ELE type. (4) Moreover, we introduce a notion of ``a growth decomposition'' for a finite string of pre-ELE type, and then give a way of constructing finite strings of pre-ELE type. As a byproduct, we show that there exist infinitely many real quadratic fields of minimal type in each period which is even and greater than or equal to 6.

Report

(5 results)
  • 2013 Annual Research Report   Final Research Report ( PDF )
  • 2012 Annual Research Report
  • 2011 Annual Research Report
  • 2010 Annual Research Report

Research Products

(10 results)

All 2014 2013 2012 2010 Other

All Journal Article (2 results) (of which Peer Reviewed: 2 results) Presentation (5 results) Remarks (3 results)

  • [Journal Article] Construction of positive integers with even period of minimal type2014

    • Author(s)
      河本史紀、岸康弘、冨田耕史
    • Journal Title

      Proc. Japan Acad. Ser. A Math. Sci.

      Volume: 90巻 Pages: 27-32

    • DOI

      10.3792/pjaa.90.27

    • NAID

      40019983593

    • Related Report
      2013 Annual Research Report 2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] Continued fractions and Gauss' class number problem for real quadratic fields2012

    • Author(s)
      河本史紀、冨田耕史
    • Journal Title

      Tokyo J. Math.

      Volume: 35巻 Pages: 213-239

    • DOI

      10.3836/tjm/1342701351

    • Related Report
      2013 Final Research Report 2012 Annual Research Report
    • Peer Reviewed
  • [Presentation] 偶数周期の連分数展開と末尾急増型主要対称部分2013

    • Author(s)
      河本史紀、岸康弘、鈴木浩志、冨田耕史
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      愛媛大学
    • Year and Date
      2013-09-23
    • Related Report
      2013 Final Research Report
  • [Presentation] 末尾急増型主要対称部分の構成法~pre-ELE型有限列の増殖変換~2013

    • Author(s)
      河本史紀、岸康弘、鈴木浩志、冨田耕史
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      愛媛大学
    • Year and Date
      2013-09-23
    • Related Report
      2013 Final Research Report
  • [Presentation] 連分数と類数1の実2次体について~ある数値実験の報告~2010

    • Author(s)
      河本史紀、冨田耕史
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      名古屋大学
    • Year and Date
      2010-09-25
    • Related Report
      2013 Final Research Report 2010 Annual Research Report
  • [Presentation] 偶数周期の連分数展開と末尾急増型主要対称部分

    • Author(s)
      河本史紀,岸康弘,鈴木浩志,冨田耕史
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      愛媛大学
    • Related Report
      2013 Annual Research Report
  • [Presentation] 末尾急増型主要対称部分の構成法~pre-ELE型有限列の増殖変換~

    • Author(s)
      河本史紀,岸康弘,鈴木浩志,冨田耕史
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      愛媛大学
    • Related Report
      2013 Annual Research Report
  • [Remarks]

    • URL

      http://auemath.aichi-edu.ac.jp/~ykishi/FSNT/10/006-kawatomi.pdf

    • Related Report
      2013 Final Research Report
  • [Remarks] 河 本 史紀、冨田 耕史、極小型実2次体と2つの有名予想、第5回福岡数論研究集会(2010)報告集、2011、45-73

    • Related Report
      2013 Final Research Report
  • [Remarks] 河本史紀・冨田耕史, 極小型実2次体と2つの有名予想, 第5回福岡数論研究集会(2010)報告集, 2011, 45-73

    • URL

      http://www.fukuoka-edu.ac.jp/~ykishi/FSNT/10/006-kawatomi.pdf

    • Related Report
      2010 Annual Research Report

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Published: 2010-08-22   Modified: 2019-07-29  

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