On the Diophantine Equations $a^x+b^y+c^z=w^2$

Abstract

Over the past decade, exponential Diophantine equations of the form ax + by = wn have been studied as if they were a phenomenon. In particular, numerous articles have focused on the cases where n = 2 or n = 4 and 2 ≤ a, b ≤ 200. However, these articles are primarily concerned with determining whether the left-hand side of the equation needs to consist of more than two
exponentials. Therefore, in this article, we investigate the exponential Diophantine equation in the form ax + by + cz = w2, using only elementary tools related to modulo concepts. We present three theorems in which the variables a, b and c vary under certain conditions, and three additional theorems where the variable c is fixed at 7. Furthermore, if we restrict our parameters a, b and c
to 2 ≤ a ≤ b ≤ c ≤ 20, then 1,330 equations have been considered. Our results confirm that 135 of these equations have been fully clarified.