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Explicit solutions to Diophantine equations attached to families of elliptic curves

Research Project

Project/Area Number 16K05079
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

FUJITA Yasutsugu  日本大学, 生産工学部, 教授 (50514163)

Project Period (FY) 2016-04-01 – 2021-03-31
Project Status Completed (Fiscal Year 2020)
Budget Amount *help
¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000)
Fiscal Year 2019: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2016: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Keywordsディオファンタスのm組 / 楕円曲線のMordell-Weil群 / 連立ペル方程式 / 楕円曲線の Mordell-Weil 群 / ディオファンタスの m 組 / 楕円曲線 / 不定方程式
Outline of Final Research Achievements

A set of m distinct positive integers is called a Diophantine m-tuple if the product of any two elements in the set increased by 1 is a perfect square. In this study, we obtain some results supporting the open conjecture asserting that any Diophantine quadruple is regular. For example, we showed that for a Diophantine triple {a,b,c} (a<b<c) the number of d's such that {a,b,c,d} (c<d) is a irregular Diophantine quadruple is at most 7.
Diophantine quadruples are closely related to elliptic curves, which are certain curves having group structures. This study also explicitly examined integral points (that is, points with integer coordinates) and generators as groups on certain families of elliptic curves.

Academic Significance and Societal Importance of the Research Achievements

任意のディオファンタスの4組は正則であると予想されており,様々な傍証はあるが,一般には正則でないディオファンタスの4組の有限性さえ示されていない.ディオファンタスの3組{a,b,c}(a<b<c)を固定してはいるが,{a,b,c,d}(c<d)が正則でないディオファンタスの4組であるようなdの個数の具体的な上限が得られた意義は大きい.一般に,与えられた1つの楕円曲線の整数点や群としての生成元を求めることは困難である.ここでは例えば u^2-v^4=m (mは0でない整数)で定義される楕円曲線の族について,階数が1または2の場合に整数点や生成元を具体的に決定した.

Report

(6 results)
  • 2020 Annual Research Report   Final Research Report ( PDF )
  • 2019 Research-status Report
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (45 results)

All 2021 2020 2019 2018 2017 2016 Other

All Int'l Joint Research (18 results) Journal Article (18 results) (of which Int'l Joint Research: 11 results,  Peer Reviewed: 17 results,  Acknowledgement Compliant: 5 results,  Open Access: 2 results) Presentation (6 results) (of which Int'l Joint Research: 2 results,  Invited: 2 results) Book (1 results) Remarks (1 results) Funded Workshop (1 results)

  • [Int'l Joint Research] The Romanian Academy(ルーマニア)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] University of Zagreb(クロアチア)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] University of Zagreb(クロアチア)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] Lingnan Normal College(中国)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] The Romanian Academy(ルーマニア)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] University of Zagreb(クロアチア)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] University of Zagreb(クロアチア)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] The Romanian Academy(ルーマニア)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] University of Zagreb(クロアチア)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] Wits University(南アフリカ)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] The Romanian Academy(ルーマニア)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] Universite de Strasbourg(フランス)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] University of Zagreb(クロアチア)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] University of the Witwatersrand(南アフリカ)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] The Rmanian Academy(ルーマニア)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] Universite de Strasbourg(フランス)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] University of Zagreb(クロアチア)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] University of the Witwatersrand(南アフリカ)

    • Related Report
      2016 Research-status Report
  • [Journal Article] Diophantine triples with largest two elements in common2021

    • Author(s)
      Mihai Cipu, Andrej Dujella and Yasutsugu Fujita
    • Journal Title

      Periodica Mathematica Hungarica

      Volume: 82 Issue: 1 Pages: 56-68

    • DOI

      10.1007/s10998-020-00331-4

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Uniqueness of Solutions to Simultaneous Pell Equations2021

    • Author(s)
      Yasutsugu Fujita and Maohua Le
    • Journal Title

      Bulletin of the Malaysian Mathematical Sciences Society

      Volume: 44 Issue: 1 Pages: 393-405

    • DOI

      10.1007/s40840-020-00959-y

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Diophantine pairs that induce certain Diophantine triples2020

    • Author(s)
      Mihai Cipu, Alan Filipin and Yasutsugu Fujita
    • Journal Title

      Journal of Number Theory

      Volume: 210 Pages: 433-475

    • DOI

      10.1016/j.jnt.2019.09.019

    • Related Report
      2020 Annual Research Report 2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Diophantine triples with largest two elements in common2020

    • Author(s)
      Mihai Cipu, Andrej Dujella and Yasutsugu Fujita
    • Journal Title

      Periodica Mathematica Hungarica

      Volume: 印刷中

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] An infinite two-parameter family of Diophantine triples2019

    • Author(s)
      Mihai Cipu, Alan Filipin and Yasutsugu Fujita
    • Journal Title

      Acta Arithmetica

      Volume: 印刷中 Issue: 1 Pages: 481-498

    • DOI

      10.1007/s40840-018-0695-9

    • Related Report
      2019 Research-status Report 2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Generators and integral points on certain quartic curves2019

    • Author(s)
      Yasutsugu Fujita and Tadahisa Nara
    • Journal Title

      Glasnik Matematicki Series III

      Volume: 54 Issue: 2 Pages: 321-343

    • DOI

      10.3336/gm.54.2.04

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] There are no Diophantine quadruples of Fibonacci numbers2018

    • Author(s)
      Yasutsugu Fuijita and Florian Luca
    • Journal Title

      Acta Arithmetica

      Volume: 185 Issue: 1 Pages: 19-39

    • DOI

      10.4064/aa170613-8-12

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] On the number of extensions of a Diophantine triple2018

    • Author(s)
      Mihai Cipu, Yasutsugu Fujita, Takafumi Miyazaki
    • Journal Title

      International Journal of Number Theory

      Volume: 14 Issue: 03 Pages: 899-917

    • DOI

      10.1142/s1793042118500549

    • Related Report
      2018 Research-status Report 2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] There are no Diophantine quadruples of Fibonacci numbers2018

    • Author(s)
      Yasutsugu Fujita and Florian Luca
    • Journal Title

      Acta Arithmetica

      Volume: 印刷中

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Two-parameter families of uniquely extendable Diophantine triples2018

    • Author(s)
      Mihai Cipu, Yasutsugu Fujita and Maurice Mignotte
    • Journal Title

      Science China Mathematics

      Volume: 61 Issue: 3 Pages: 421-438

    • DOI

      10.1007/s11425-015-0638-0

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] The Mordell--Weil bases for the elliptic curve of the form y^2=x^3-m^2x+n^22018

    • Author(s)
      Yasutsugu Fujita and Tadahisa Nara
    • Journal Title

      Publicationes Mathematicae Debrecen

      Volume: 92 Issue: 1-2 Pages: 79-99

    • DOI

      10.5486/pmd.2018.7719

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] The regularity of Diophantine quadruples2017

    • Author(s)
      Yasutsugu Fujita, Takafumi Miyazaki
    • Journal Title

      Transactions of the American Mathematical Society

      Volume: 印刷中 Issue: 6 Pages: 3803-3831

    • DOI

      10.1090/tran/7069

    • Related Report
      2018 Research-status Report 2017 Research-status Report 2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] On Diophantine quadruples of Fibonacci numbers2017

    • Author(s)
      Yasutsugu Fujita and Florian Luca
    • Journal Title

      Glasnik Matematicki Series III

      Volume: 52 Issue: 2 Pages: 221-234

    • DOI

      10.3336/gm.52.2.02

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] ディオファントスのm組に関する最近の進展2017

    • Author(s)
      藤田育嗣
    • Journal Title

      日本大学生産工学部研究報告A

      Volume: 50 Pages: 71-74

    • NAID

      40021254133

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Mordell-Weil bases for the elliptic curve of the form y^2=x^3-m^2x+n^22017

    • Author(s)
      Yasutsugtu Fujita and Tadahisa Nara
    • Journal Title

      Publicationes Mathematicae Debrecen

      Volume: 印刷中

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Two-parameter families of uniquely extendable Diophantine triples2017

    • Author(s)
      Mihai Cipu, Yasutsugu Fujita and Maurice Mignotte
    • Journal Title

      Science China Mathematics

      Volume: 印刷中

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] ディオファントスの m 組に関する最近の進展2017

    • Author(s)
      藤田育嗣
    • Journal Title

      日本大学生産工学部研究報告A

      Volume: 印刷中

    • NAID

      40021254133

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Diophantus の3 組の4 組への拡張可能性2017

    • Author(s)
      藤田育嗣,宮崎隆史
    • Journal Title

      RIMS 研究集会報告集

      Volume: - Pages: 111-123

    • Related Report
      2016 Research-status Report
    • Open Access / Acknowledgement Compliant
  • [Presentation] 非正則なディオファンタスの 4 組の個数の評価2020

    • Author(s)
      藤田育嗣
    • Organizer
      2020大分整数論研究集会
    • Related Report
      2020 Annual Research Report
  • [Presentation] The regularity of extensions of Diophantine triples or pairs2019

    • Author(s)
      Yasutsugu Fujita
    • Organizer
      Representation Theory XVI
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Non-existence of Diophantine quadruples consisting of Fibonacci numbers2019

    • Author(s)
      Yasutsugu Fujita
    • Organizer
      Diophantine Analysis and Related Fields 2019
    • Related Report
      2018 Research-status Report
  • [Presentation] Simultaneous Diophantine triples and quadruples2018

    • Author(s)
      Yasutsugu Fujita
    • Organizer
      Conference on Diophantine m-tuples and Related Problems II
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Bounds for irregular Diophantine quadruples2017

    • Author(s)
      Yasutsugu Fujita
    • Organizer
      Diophantine Analysis and Related Fields 2017
    • Place of Presentation
      日本大学理工学部 駿河台キャンパス お茶の水校舎
    • Related Report
      2016 Research-status Report
  • [Presentation] The unique extensions of two parametric families of Diophantine triples2016

    • Author(s)
      Yasutsugu Fujita
    • Organizer
      Problems and prospects in Analytic Number Theory
    • Place of Presentation
      京都大学数理解析研究所
    • Related Report
      2016 Research-status Report
  • [Book] 解析的整数論とその周辺 数理解析研究所講究録 18132018

    • Author(s)
      門田慎也 他 (藤田育嗣 編)
    • Total Pages
      225
    • Publisher
      数理解析研究所
    • Related Report
      2018 Research-status Report
  • [Remarks] 2017年度RIMS共同研究(公開型)「解析的整数論とその周辺」

    • URL

      https://sites.google.com/a/nihon-u.ac.jp/yasutsugu-fujita-s-homepage/2017nian-durims-yan-jiu-ji-hui-jie-xi-de-zheng-shu-luntosono-zhou-bian

    • Related Report
      2017 Research-status Report
  • [Funded Workshop] RIMS Workshop 2017 "Analytic Number Theory and Related Areas"2017

    • Related Report
      2017 Research-status Report

URL: 

Published: 2016-04-21   Modified: 2022-02-22  

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