A note on the ternary purely exponential Diophantine equation $A^x+B^y=C^z$ with $A+B=C^2$

Abstract

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 $(x,y,z)=(1,1,2)$.

5 pages, accepted for publication (2020)