A note on the exponential Diophantine equation ((A^2n)^x+(B^2n)^y=((A^2+B^2)n)^z)

Let (A) $B$ be positive integers such that $min{A,B}>1$, $gcd(A,B) = 1$ and $2|B.$ In this paper, using an upper bound for solutions of ternary purely exponential Diophantine equations due to R. Scott and R. Styer, we prove that, for any positive integer $n$, if $A >B^3/8$, then the equation $(A^2 n)^x + (B^2 n)^y = ((A^2 + B^2)n)^z$ has no positive integer solutions $(x,y,z)$ with $x > z > y$; if $B>A^3/6$, then it has no solutions $(x,y,z)$ with $y>z>x$. Thus, combining the above conclusion with some existing results, we can deduce that, for any positive integer $n$, if $Bequiv 2 pmod{4}$ and $A >B^3/8$, then this equation has only the positive integer solution $(x,y,z)=(1,1,1)$