A Discussion on the Solution(s) of the Diophantine Equation 3^x + 15^y =Z²

Abstract

The absence of a generalized method for solving Diophantine equations having more unknowns than a number of equations is a challenge for researchers in different fields. The presence of the Diophantine equation is reported in the study of the Hydrogen spectrum, quantum Hall effect, chemistry, cryptography, etc. Some special types of Diophantine equations could be addressed with the help of Catalan’s conjecture and Congruence theory. The Diophantine equation 3^x + 15^y =z² is addressed in this paper to find the solution(s) in positive integers. It is found that the equation has only two solutions of (x,y,z) as  (1,0,2) and (0,1,4) in non-negative integers.