A note on the exponential Diophantine equation $(4m^2+1)^x+(5m^2-1)^y=(3m)^z$
Volume 139 / 2015
Colloquium Mathematicum 139 (2015), 121-126
MSC: 11D61.
DOI: 10.4064/cm139-1-7
Abstract
Let $m$ be a positive integer. Using an upper bound for the solutions of generalized Ramanujan–Nagell equations given by Y. Bugeaud and T. N. Shorey, we prove that if $3\nmid m$, then the equation $(4m^2+1)^x+(5m^2-1)^y=(3m)^z$ has only the positive integer solution $(x,y,z)=(1,1,2)$.
