Solution of the Exponential Non-Linier Diophantine Equations 3^x+5^y=z^2

Abstract

This research will find a solution (if any) from the exponential non-linear  Diophantine equation  using the basic theory of congruence. Sroysang (2012) has examined the solution of the Exponential non-linear  Diophantine equation , with integers. The solution obtained is ( ) = (1,0,2). Sroysang used the Catalan Conjecture to determine the solution. In this paper we will also find a solution of the exponential  non- linear Diophantine equation  with a fundamental approach in the theory of Numbers, namely the theory of congruence. In General If we are given a  non-linear Diophantine  equation there are 3 possibilities for finding a solution of the non-linear Diophantine equation that is a single solution, many solutions, or no solution. The research methodology in this paper is a journal review and is linked to theory of congruence. It can be proven that the non-linear Diophantine equation  has a single solution (x, y, z) = (1,0,2) for non-negative integers.