Czechoslovak Mathematical Journal, online first, 9 pp.

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

Xiaoying Du

Received November 28, 2015.   First published July 11, 2017.

Xiaoying Du, School of Mathematics, Jinzhong University, Wenhua Street 199, Yuci Qu, Jinzhong 030619, Shanxi, P. R. China, duxiaoying83@163.com

Abstract: 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 $(x,y,z)$.

Keywords: exponential diophantine equation; upper bound for solutions; singular number

Classification (MSC 2010): 11D61

DOI: 10.21136/CMJ.2017.0645-15

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