  Let $\ell, m, r$ be fixed positive integers such that $2\nmid \ell$, $3\nmid
\ell m$, $\ell>r$ and $3\mid r$. In this paper, using the BHV theorem on the
existence of primitive divisors of Lehmer numbers, we prove that if
$\min\{r\ell m^2-1,(\ell-r)\ell m^2+1\}>30$, then the equation $(r\ell
m^2-1)^x+((\ell -r)\ell m^2+1)^y=(\ell m)^z$ has only the positive integer
solution $(x,y,z)=(1,1,2)$.

Number Theory