On the diophantine equation $x^y-y^x=c^z$
Tom 128 / 2012
Colloquium Mathematicum 128 (2012), 277-285
MSC: Primary 11D61; Secondary 11D41.
DOI: 10.4064/cm128-2-13
Streszczenie
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 ${\rm gcd}(x,y)=1$ then $(x,y,z)=(2,1,k)$, $(3, 2, k)$, $k\geq 1$ if $c=1$, and either $(x,y,z)=(c^k+1,1,k)$, $k\geq 1$ or $2\leq x < y\leq\max\{1.5\times 10^{10}, c\}$ if $c\geq 2$.
