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On the distribution of the residual order/index of the residual class a(mod n) in the residual group with composite moduli.

Research Project

Project/Area Number 18540054
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionMeiji Gakuin University

Principal Investigator

MURATA Leo  Meiji Gakuin University, 経済学部, 教授 (30157789)

Co-Investigator(Kenkyū-buntansha) 北岡 良之  名城大学, 理工学部, 教授 (40022686)
岡崎 龍太郎  同志社大学, 工学部, 専任講師 (20268113)
知念 宏司  近畿大学, 理工学部, 准教授 (30419486)
Project Period (FY) 2006 – 2008
Project Status Completed (Fiscal Year 2008)
Budget Amount *help
¥3,340,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥540,000)
Fiscal Year 2008: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2007: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2006: ¥1,000,000 (Direct Cost: ¥1,000,000)
Keywords剰余位数 / 剰余指数 / 原始根 / Code理論 / 剩余位数 / Gray Code / Sum of digits関数 / 原始根に関するArtin予想 / 原始根に関するArtinの予想
Research Abstract

素数pを法とする剰余類群は、位数p-1の単純な巡回群になる。自然数aを固定し、素数pを動かして剰余類a(mod p)の位数の分布を調べる問題は、すでに村田-知念によってほぼ解決している。ところが二つの異なる素数の積pqを法とする剰余類群は、群構造が複雑になり、例えば原始根の分布問題等は、素数pの剰余類群の場合と大きく異なった様相を呈する。今回は剰余類a(mod pq)の位数の分布をp,qを動かした場合に調べ、この問題の場合、現象としては素数pを法とする場合と大きな違いはなさそうだという結論を得た。

Report

(4 results)
  • 2008 Annual Research Report   Final Research Report ( PDF )
  • 2007 Annual Research Report
  • 2006 Annual Research Report

Research Products

(17 results)

All 2009 2008 2006 Other

All Journal Article (14 results) (of which Peer Reviewed: 11 results) Presentation (3 results)

  • [Journal Article] A statistical relation of roots of a polynomial in different local fields2009

    • Author(s)
      Y. Kitaoka
    • Journal Title

      Mathematics of Computation

      Pages: 523-536

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] A statistical relation of roots of a polynomial in different local fields2009

    • Author(s)
      Y. Kitaoka
    • Journal Title

      Mathematics of Computation 78

      Pages: 523-536

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the average of some q-additive functions2008

    • Author(s)
      Leo Murata, Yuichi Kamiya
    • Journal Title

      Annales Univ. Sci. Budapest., Sect. Comp. Vol. 29

      Pages: 39-63

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] An abundance of invariant polynomials satisfying the Riemann hypothesis2008

    • Author(s)
      K. Chinen
    • Journal Title

      Discrete Math. Vol. 308

      Pages: 6426-6440

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] An abundance of invariant polynomials satisfying the Riemann hypothesis.2008

    • Author(s)
      K. Chinen
    • Journal Title

      Discrete Math. 308

      Pages: 6426-6440

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the average of some q-additive functions.2008

    • Author(s)
      Leo Murata and Yuichi Kamiya
    • Journal Title

      Annales Univ. Sci. Budapest., Sect, Comp. 29

      Pages: 39-63

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On a distribution property of the residual order of a (mod p), part III2006

    • Author(s)
      L. Murata, K. Chinen
    • Journal Title

      J. of Math. Soc. Japan Vol. 58-3

      Pages: 693-720

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] On a distribution property of the residual order of a (mod p), part IV,2006

    • Author(s)
      L. Murata, K. Chinen
    • Journal Title

      "Number Theory: Tradition and Modernization", Edited by Y. Tanigawa and W. ZhangSpringer Science

      Pages: 11-22

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] Densities of sets of primes related to decimal expansion of rational numbers2006

    • Author(s)
      T. Hadano, Y. Kitaoka, T. Kubota, M. Nozaki
    • Journal Title

      "Number Theory: Tradition and Modernization", Edited by Y. Tanigawa and W. Zhang, Springer Science

      Pages: 67-80

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] On a distribution property of the residual order of a(mod p) - III2006

    • Author(s)
      Leo Murata (with K. Chinen)
    • Journal Title

      J. of Math. Soc. Japan 58-3

      Pages: 693-720

    • NAID

      10018381051

    • Related Report
      2006 Annual Research Report
  • [Journal Article] ON A DISTRIBUTION PROPERTY OF THE RESIDUAL ORDER OF a(mod p) - IV2006

    • Author(s)
      Leo Murata (with K. Chinen)
    • Journal Title

      Number Theory : Tradition and Modernization (Edited by Y. Tanigawa, W. Zhang) (Springer Science)

      Pages: 11-22

    • Related Report
      2006 Annual Research Report
  • [Journal Article] DENSITIES OF SETS OF PRIMES RELATED TO DECIMAL EXPANSION OF RATIONAL NUMBERS2006

    • Author(s)
      Toshihiro Hadano, Yoshiyuki Kitaoka, tomio Kubota, Michihiro Nozaki
    • Journal Title

      Number Theory : Tradition and Modernization (Edited by Y. Tanigawa, W. Zhang) (Springer Science)

      Pages: 67-80

    • Related Report
      2006 Annual Research Report
  • [Journal Article] A statistical relation of roots of a polynomial in different local fields

    • Author(s)
      Ybshiyuki Kitaoka
    • Journal Title

      Mathematics of Computation 掲載確定

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] An abundance of invariant polynomials satisfying the Riemann hypothesis

    • Author(s)
      Koji Chinen
    • Journal Title

      Discrete Math. 掲載確定

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Presentation] A statistical relation of roots of a polynomial in different local fields2008

    • Author(s)
      北岡 良之
    • Organizer
      The 5^<th> Japan-China Seminar on Number Theory
    • Place of Presentation
      近畿大学・理工学部
    • Year and Date
      2008-08-30
    • Related Report
      2008 Annual Research Report 2008 Final Research Report
  • [Presentation] Some q-additive functions and related topics2008

    • Author(s)
      村田 玲音
    • Organizer
      The 5^<th> Japan-China Seminar on Number Theory
    • Place of Presentation
      近畿大学・理工学部
    • Year and Date
      2008-08-28
    • Related Report
      2008 Annual Research Report 2008 Final Research Report
  • [Presentation] On a property of the multiplicative order of a(mod p)2006

    • Author(s)
      村田 玲音
    • Organizer
      The 4^<th> Japan-China Seminar on Number Theory
    • Place of Presentation
      中国・山東大学
    • Year and Date
      2006-09-03
    • Related Report
      2008 Final Research Report

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Published: 2006-03-31   Modified: 2016-04-21  

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