2022 Fiscal Year Final Research Report
Study on exponential Diophantine equations related to Jesmanowicz' conjecture
| Project/Area Number |
18K03247
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 11010:Algebra-related
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| Research Institution | Oita University |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2023-03-31
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| Keywords | Jesmanowicz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 |
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| 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.
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| Free Research Field |
指数型不定方程式
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| 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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