Exploring 8^x + n^y = z² through Associated Elliptic Curves

Abstract

This paper investigates the exponential Diophantine equation 8x+ ny = z2, where n > 1 is an odd positive integer. We characterize solutions for the base cases (x = 0 or y = 0) and describe, based on implications of Bennett and Skinner’s theorem, that no solutions exist for y > 2 in certain cases. For y = 1 and y = 2, we employ elliptic curve methods, focusing on the equations z2 = t3+n and z2 = t3+n2, where t = 2x. This work generalizes known results for specific cases and provides insights into this class of Diophantine equations and their associated elliptic curves.