| Project/Area Number |
18K03247
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Multi-year Fund |
|---|
| Section | 一般 |
|---|
| Review Section |
Basic Section 11010:Algebra-related
|
|---|
| Research Institution | Oita University |
|---|
Principal Investigator |
|
|---|
| Project Period (FY) |
2018-04-01 – 2023-03-31
|
| Project Status |
Completed (Fiscal Year 2022)
|
| Budget Amount *help |
¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000)
Fiscal Year 2020: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2019: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2018: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
|
|---|
| 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 |
|---|
| 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.
|
| 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.
|
