On the Unsolvability of the Exponential Diophantine Equation \( 23^x + 22^y = z^2 \) in Nonnegative Integers

Abstract

We prove that the exponential Diophantine equation \( 23^x + 22^y = z^2 \) has no nonnegative integer solutions. Using a combination of modular arithmetic, parity analysis, and computational verification, we demonstrate that the equation leads to contradictions under all possible cases. Our work extends previous results on equations of the form \( p^x + (p-1)^y = z^2 \) and highlights the interplay between theoretical and computational methods in solving Diophantine problems. We also provide computational data for small values of \( x \) and \( y \) to support our theoretical findings. Furthermore, we generalize our approach to other equations of the form \( w^x + (w-1)^y = z^2 \), where \( w \) is a positive integer of a specific form. This study contributes to the broader understanding of exponential Diophantine equations and their solutions.