The Exponential Diophantine Equation 4m2+1x+5m2-1y=(3m)z

Abstract

Let m be a positive integer. In this paper, using some properties of exponential diophantine equations and some results on the existence of primitive divisors of Lucas numbers, we prove that if $m>90$ and $3|m$, then the equation ${\left(4m^2+1\right)}^x\hspace{0.17em}+\hspace{0.17em}{\left({5m}^2-1\right)}^y={(3m)}^z$ has only the positive integer solution $(x,y,z)=(1,1,2)$.