On the Diophantine Equation $4^x-p^y=3z^2$ where $p$ is a Prime

Julius Fergy Tiongson Rabago

Abstract

We find all solutions to $4^x-7^y =z^2$ and $4^x-11^y =z^2$ to complement the results found by Suvarnamani, et. al. in [1]. We also consider the two Diophantine equations $4^x - 7^y = 3 z^2$ and $4^x - 19^y = 3 z^2$ and show that these two equations have exactly two solutions $(x,y,z)$ in non-negative integers, i.e. $(x,y,z)\in\{(0,0,0), (1,0,1)\}$. In fact, the Diophantine equation $4^x-p^y=3z^2$ has the two solutions $(0,0,0)$ and $(1,0,1)$ under some additional assumption on $p$. These results were all obtained using elementary methods and Mih\u{a}ilescu's Theorem.Finally, we end our paper with an open problem.