On the exponential Diophantine equation $p\cdot 3^x+p^y=z^2$ with $p$ a prime number

Abstract

In this paper we find non-negative integer solutions for exponential Diophantine equations of the type $p\cdot 3^x+p^y=z^2,$ where $p$ is a prime number. We prove that such equation has a unique solution $(x,y,z)=\left({\log }_3(p-2),0,p-1\right)$ if $2\ne p\equiv 2(\mathrm{mod}3)$ and $(x,y,z)=(0,1,2)$ if $p=2$. We also display the infinite solution set of that equation in the case $p=3$. Finally, a brief discussion of the case $p\equiv 1(\mathrm{mod}3)$ is made, where we display an equation that does not have a non-negative integer solution and leave some open questions. The proofs are based on the use of the properties of the modular arithmetic.