On Diophantine Equations \(2^x + 3y^2 = 4^z\) and \( 2^x + 7y^2 = 4^z\)

Abstract

In this note, we study the Diophantine equations \(2^x + 3y^2 = 4^z\) and \(2^x+7y^2 = 4^z\). Also, we give solutions to \(2^x+dy^2 = 4^z\) for \(d = (2^k-1)/9, k\) a natural number = 0 (mod 6).