On the Diophantine Equation $p^x + (p + 5k)^y = z^2$ (European Journal of Pure and Applied Mathematics, 2025)
With the use of modular arithmetic and other fundamental number theoretic methods, as well as the concepts of floor function and the principle of mathematical induction, this study searches for possible nonnegative integer solutions of exponential Diophantine equations of the form $p^x + (p+5k)^y = z^2$, where $k \in \mathbb{N}$. Results are obtained for the following cases: \begin{itemize}\item [a)] when $p =2$; or\item [b)] when $p$ and $p + 5k$ are prime pairs.In addition, the study is limited only to solutions where $x$ and $y$ are not both greater than 1.