A Note on the Exponential Diophantine Equation \((44m^2+1)^x+(5m^2-1)^y=(7m)^z\)

Let $m$ be a positive integer. We show that the exponential Diophantine equation $(44m^2+1)^x+(5m^2-1)^y=(7m)^z$ has only the positive integer solution $(x,y,z)=(1,1,2)$ under some conditions. The proof is based on elementary methods, Baker’s method and linear forms in $p$-adic logarithms.