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

Xiaoying Du

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 3, page 645-653
  • ISSN: 0011-4642

Abstract

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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 - D v 2 = 1 , and let h ( 4 D ) denote the class number of binary quadratic primitive forms of discriminant 4 D . If D satisfies 2 ∤ D and v 1 h ( 4 D ) ≡ 0 ( mod 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 ∣ 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 ) .

How to cite

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Du, Xiaoying. "On the exponential diophantine equation $x^y+y^x=z^z$." Czechoslovak Mathematical Journal 67.3 (2017): 645-653. <http://eudml.org/doc/294089>.

@article{Du2017,
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 $v_1h(4D)\equiv 0 \hspace\{4.44443pt\}(\@mod \; D)$, 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 $\max \lbrace x,y,z\rbrace <480000$ and both $x$, $y$ are singular numbers. Thus, one can possibly prove that the equation has no positive integer solutions $(x,y,z)$.},
author = {Du, Xiaoying},
journal = {Czechoslovak Mathematical Journal},
keywords = {exponential diophantine equation; upper bound for solutions; singular number},
language = {eng},
number = {3},
pages = {645-653},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the exponential diophantine equation $x^y+y^x=z^z$},
url = {http://eudml.org/doc/294089},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Du, Xiaoying
TI - On the exponential diophantine equation $x^y+y^x=z^z$
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 3
SP - 645
EP - 653
AB - 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 $v_1h(4D)\equiv 0 \hspace{4.44443pt}(\@mod \; D)$, 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 $\max \lbrace x,y,z\rbrace <480000$ and both $x$, $y$ are singular numbers. Thus, one can possibly prove that the equation has no positive integer solutions $(x,y,z)$.
LA - eng
KW - exponential diophantine equation; upper bound for solutions; singular number
UR - http://eudml.org/doc/294089
ER -

References

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