On the diophantine equation $x^y-y^x=c^z$

Zhongfeng Zhang, Jiagui Luo, and Pingzhi Yuan. "On the diophantine equation $x^y - y^x = c^z$." Colloquium Mathematicae 128.2 (2012): 277-285. <http://eudml.org/doc/283734>.

@article{ZhongfengZhang2012,
abstract = {Applying results on linear forms in p-adic logarithms, we prove that if (x,y,z) is a positive integer solution to the equation $x^y - y^x = c^z$ with gcd(x,y) = 1 then (x,y,z) = (2,1,k), (3,2,k), k ≥ 1 if c = 1, and either $(x,y,z) = (c^k+1,1,k)$, k ≥ 1 or $2 ≤ x < y ≤ max\{1.5×10^\{10\},c\}$ if c ≥ 2.},
author = {Zhongfeng Zhang, Jiagui Luo, Pingzhi Yuan},
journal = {Colloquium Mathematicae},
keywords = {exponential Diophantine equation; linear forms in logarithms},
language = {eng},
number = {2},
pages = {277-285},
title = {On the diophantine equation $x^y - y^x = c^z$},
url = {http://eudml.org/doc/283734},
volume = {128},
year = {2012},
}

TY - JOUR
AU - Zhongfeng Zhang
AU - Jiagui Luo
AU - Pingzhi Yuan
TI - On the diophantine equation $x^y - y^x = c^z$
JO - Colloquium Mathematicae
PY - 2012
VL - 128
IS - 2
SP - 277
EP - 285
AB - Applying results on linear forms in p-adic logarithms, we prove that if (x,y,z) is a positive integer solution to the equation $x^y - y^x = c^z$ with gcd(x,y) = 1 then (x,y,z) = (2,1,k), (3,2,k), k ≥ 1 if c = 1, and either $(x,y,z) = (c^k+1,1,k)$, k ≥ 1 or $2 ≤ x < y ≤ max{1.5×10^{10},c}$ if c ≥ 2.
LA - eng
KW - exponential Diophantine equation; linear forms in logarithms
UR - http://eudml.org/doc/283734
ER -