An application of the BHV theorem to a new conjecture on exponential diophantine equations

Abstract

Let $A$, $B$ be fixed positive integers such that $\min \{A,B\}>1$, $\gcd (A,B)=1$ and $AB\equiv 0\mathrm{mod}2$, and let $n$ be a positive integer with $n>1$. In this paper, using a deep result on the existence of primitive divisors of Lucas numbers due to Y. Bilu, G. Hanrot and P. M. Voutier \cite{BHV}, we prove that if $A>8B^3$, then the equation $(\ast )(A^{2}n{)}^x+(B^{2}n{)}^y=((A^2+B^2)n{)}^z$ has no positive integer solutions $(x,y,z)$ with $x>z>y$. Combining the above conclusion with some existing results, we can deduce that if $A>8B^3$ and $B\equiv 2\mathrm{mod}4$, then (*) has only the positive integer solution $(x,y,z)=(1,1,1)$.