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Study on integer solutions of exponential Diophantine equations

Research Project

Project/Area Number 15K04786
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionOita University

Principal Investigator

Terai Nobuhiro  大分大学, 理工学部, 教授 (00236978)

Project Period (FY) 2015-04-01 – 2018-03-31
Project Status Completed (Fiscal Year 2017)
Budget Amount *help
¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2015: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Keywords指数型不定方程式 / 整数解 / Jesmanowicz予想 / Terai予想 / Baker理論 / 一般化されたFermat方程式 / 楕円曲線 / 数論 / 不定方程式
Outline of Final Research Achievements

Our purpose of this research is to determine integer solutions of exponential Diophantine equations (1) a^x + b^y = c^z, (2) (pm^2 + 1)^x + (qm^2 - 1)^y = (rm)^z with p+q=r^2 (3) (a^φ(m)-1)/m = x^l, where φ is Euler's totient function, under some conditions. Our method is based on elementary methods, Baker's method and deep results on generalized Fermat equations via sophisticated arguments in the theory of elliptic curves and modular forms.

Report

(4 results)
  • 2017 Annual Research Report   Final Research Report ( PDF )
  • 2016 Research-status Report
  • 2015 Research-status Report
  • Research Products

    (17 results)

All 2018 2017 2016 2015

All Journal Article (5 results) (of which Peer Reviewed: 5 results,  Open Access: 2 results,  Acknowledgement Compliant: 2 results) Presentation (9 results) Funded Workshop (3 results)

  • [Journal Article] On the exponential Diophantine equation (3pm^2-1)^x+(p(p-3)m^2+1)^y=(pm)^z2017

    • Author(s)
      Nobuhiro Terai, Takeshi Hibino
    • Journal Title

      Period. Math. Hungar.

      Volume: 74 Issue: 2 Pages: 227-234

    • DOI

      10.1007/s10998-016-0162-z

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the exponential Diophantine equation a^x+lb^y=c^z2016

    • Author(s)
      Nobuhiro Terai, Takeshi Hibino
    • Journal Title

      Int. J. Algebra

      Volume: 10 Pages: 393-403

    • DOI

      10.12988/ija.2016.6747

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] On exponential Diophantine equations containing the Euler quotient2015

    • Author(s)
      Nobuhiro Terai
    • Journal Title

      Bull. Australian Math. Soc.

      Volume: 91 Issue: 1 Pages: 11-18

    • DOI

      10.1017/s0004972714000719

    • Related Report
      2015 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the exponential Diophantine equation (12m^2 + 1)^x + (13m^2-1)^y = (5m)^z2015

    • Author(s)
      Nohuhiro Terai, Takeshi Hibino
    • Journal Title

      International Journal of Algebra

      Volume: 9 Pages: 261-272

    • DOI

      10.12988/ija.2015.5529

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] On Jesmanowicz' conjecture concerning primitive Pythagorean triples II2015

    • Author(s)
      Nobuhiro Terai, Takafumi Miyazaki
    • Journal Title

      Acta Math. Hungarica

      Volume: 147 Issue: 2 Pages: 286-293

    • DOI

      10.1007/s10474-015-0552-3

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Presentation] 一般化されたRamanujan-Nagell方程式 x^2+(2c-1)^m=c^n について2018

    • Author(s)
      寺井伸浩
    • Organizer
      第138回日本数学会九州支部例会
    • Related Report
      2017 Annual Research Report
  • [Presentation] 指数型不定方程式 a^x+lb^y=c^z について2017

    • Author(s)
      寺井伸浩
    • Organizer
      第136回日本数学会九州支部例会
    • Place of Presentation
      福岡教育大学(福岡県福岡市)
    • Year and Date
      2017-02-18
    • Related Report
      2016 Research-status Report
  • [Presentation] On the exponential Diophantine equation a^x + b^y = c^z2017

    • Author(s)
      寺井伸浩
    • Organizer
      第11回福岡数論研究集会
    • Related Report
      2017 Annual Research Report
  • [Presentation] 指数型不定方程式 a^x + 3b^y = c^z について2016

    • Author(s)
      寺井伸浩
    • Organizer
      第135回日本数学会九州支部例会
    • Place of Presentation
      長崎大学(長崎県長崎市)
    • Year and Date
      2016-10-22
    • Related Report
      2016 Research-status Report
  • [Presentation] Euler商に関する指数型不定方程式について2016

    • Author(s)
      寺井伸浩
    • Organizer
      第10回福岡数論研究集会
    • Place of Presentation
      九州大学伊都キャンパス(福岡県福岡市)
    • Year and Date
      2016-08-09
    • Related Report
      2016 Research-status Report
  • [Presentation] 指数型不定方程式 a^x+b^y=c^z の解の個数について2016

    • Author(s)
      寺井伸浩
    • Organizer
      愛知数論セミナー
    • Place of Presentation
      愛知工業大学本山キャンパス(愛知県名古屋市)
    • Year and Date
      2016-05-21
    • Related Report
      2016 Research-status Report
  • [Presentation] 指数型不定方程式に関する Jesmanowicz 予想の一般化について2015

    • Author(s)
      寺井伸浩
    • Organizer
      第133回日本数学会九州支部例会
    • Place of Presentation
      佐賀大学理工学部 (佐賀県佐賀市)
    • Year and Date
      2015-10-24
    • Related Report
      2015 Research-status Report
  • [Presentation] 指数型不定方程式 a^x+b^y=c^z について2015

    • Author(s)
      寺井伸浩
    • Organizer
      北陸数論セミナー
    • Place of Presentation
      金沢大学サテライトプラザ (石川県金沢市)
    • Year and Date
      2015-09-17
    • Related Report
      2015 Research-status Report
  • [Presentation] 指数型不定方程式 (pm^2+1)^x+(qm^2-1)^y=(rm)^z について2015

    • Author(s)
      寺井伸浩
    • Organizer
      早稲田大学整数論セミナー
    • Place of Presentation
      早稲田大学西早稲田キャンパス (東京都新宿区)
    • Year and Date
      2015-05-15
    • Related Report
      2015 Research-status Report
  • [Funded Workshop] 2017大分熊本整数論研究集会2017

    • Related Report
      2017 Annual Research Report
  • [Funded Workshop] 2016大分整数論研究集会2016

    • Place of Presentation
      ホルトホール大分サテライトキャンパス(大分県大分市)
    • Year and Date
      2016-10-08
    • Related Report
      2016 Research-status Report
  • [Funded Workshop] 2015大分整数論研究集会2015

    • Place of Presentation
      ホルトホール大分サテライトキャンパス (大分県大分市)
    • Year and Date
      2015-09-01
    • Related Report
      2015 Research-status Report

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Published: 2015-04-16   Modified: 2019-03-29  

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