| Title:
|
On the exponential diophantine equation $x^y+y^x=z^z$ (English) |
| Author:
|
Du, Xiaoying |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
67 |
| Issue:
|
3 |
| Year:
|
2017 |
| Pages:
|
645-653 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
For any positive integer $D$ which is not a square, let $(u_1,v_1)$ be the least positive integer solution of the Pell equation $u^2-Dv^2=1,$ and let $h(4D)$ denote the class number of binary quadratic primitive forms of discriminant $4D$. If $D$ satisfies $2\nmid D$ and $v_1h(4D)\equiv 0 \pmod D$, then $D$ is called a singular number. In this paper, we prove that if $(x,y,z)$ is a positive integer solution of the equation $x^y+y^x=z^z$ with $2\mid z$, then maximum $\max \{x,y,z\}<480000$ and both $x$, $y$ are singular numbers. Thus, one can possibly prove that the equation has no positive integer solutions $(x,y,z)$. (English) |
| Keyword:
|
exponential diophantine equation |
| Keyword:
|
upper bound for solutions |
| Keyword:
|
singular number |
| MSC:
|
11D61 |
| idZBL:
|
Zbl 06770122 |
| idMR:
|
MR3697908 |
| DOI:
|
10.21136/CMJ.2017.0645-15 |
| . |
| Date available:
|
2017-09-01T12:20:57Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146851 |
| . |
| Reference:
|
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