Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

HOME ALL ARTICLES View

Bull. Korean Math. Soc. 2015; 52(4): 1133-1138

Printed July 31, 2015

https://doi.org/10.4134/BKMS.2015.52.4.1133

Copyright © The Korean Mathematical Society.

On the Diophantine equation $(an)^{x}+(bn)^{y}=(cn)^{z}$

Mi-Mi Ma and Jian-Dong Wu

Nanjing Normal University, Nanjing Normal University

Abstract

In 1956, Je\'{s}manowicz conjectured that, for any positive integer $n$ and any primitive Pythagorean triple $(a,b,c)$ with $a^2+b^2=c^2$, the equation $(an)^x+(bn)^y=(cn)^z$ has the unique solution $(x,y,z)=(2,2,2)$. In this paper, under some conditions, we prove the conjecture for the primitive Pythagorean triples $(a,b,c)=(4k^{2}-1,4k,4k^{2}+1)$.

Keywords: Je\'{s}manowicz' conjecture, Diophantine equation, Pythagorean triple

MSC numbers: 11D61