On the exponential diophantine equation $x^y+y^x=z^z$

Title:

On the exponential diophantine equation $x^y+y^x=z^z$

Creator:

Du, Xiaoying

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:1d141f40-31f3-40c5-ac67-2447a14af5c2
uuid:1d141f40-31f3-40c5-ac67-2447a14af5c2
issn:0011-4642

Subject:

matematika, mathematics, exponential diophantine equation, upper bound for solutions, singular number, 13, and 51

Type:

model:article and TEXT

Format:

print, bez média, and svazek

Description:

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 $v_1h(4D)\equiv0 \pmod D$, 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 $(x,y,z)$., Xiaoying Du., and Obsahuje bibliografické odkazy

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

645-653

Source:

Czechoslovak Mathematical Journal | 2017 Volume:67 | Number:3

Harvested from:

CDK

Metadata only:

false