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2022 Fiscal Year Final Research Report

Study on exponential Diophantine equations related to Jesmanowicz' conjecture

Research Project

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Project/Area Number 18K03247
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 11010:Algebra-related
Research InstitutionOita University

Principal Investigator

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

Project Period (FY) 2018-04-01 – 2023-03-31
KeywordsJesmanowicz予想 / 指数型不定方程式 / Ramanujan-Nagell方程式 / 一般化されたFermat方程式 / 整数解 / Baker理論 / 楕円曲線
Outline of Final Research Achievements

Our purpose of this research is to determine all integer solutions of the following three exponential Diophantine equations: (1) a^x + b^y = c^z with
a^2+b^2=c^2, (2) a^x + b^y = c^z with a^p+b^q=c^r, (3) x^2+b^m=c^n with a^2+b^2=c^2 and b even. Our strategy is based on elementary methods, Baker theory, and deep results on generalized Ramanujan-Nagell equations and Fermat equations.

Free Research Field

指数型不定方程式

Academic Significance and Societal Importance of the Research Achievements

Jesmanowicz予想と関係する指数型不定方程式 a^x + b^y = c^z(ここでa^p + b^q = c^r)や一般化されたRamanujan-Nagell方程式 x^2+b^m=c^n (ここでa^2+b^2=c^r)の整数解について, いくつかの条件の下でいろいろな場合に決定することができた. また, 類数・線形数列・楕円曲線を用いて, 指数型不定方程式の整数解に関する興味深い予想を提示でき, 今後の指数型不定方程式の研究に有意義となるものである.

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Published: 2024-01-30  

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