Solution to Non-Linear Exponential Diophantine Equation 13 x + 31 y = z 2

Abstract

Linear Diophantine Equation is a polynomial equation with degree 1 and non-zero integer coefficient. The general form of Diophantine Linear equation with 2 variables is ax + by = c with a, b, c ϵ Z and a, b ≠ 0 . This may be stated as congruency ax ≡ b ( mod m) . Therefore, Diophantine Linear equation ax + by = c may be solved if and only if the equivalent of congruency ax ≡ b (mod m ) may be solved. If the Linear Diophantine Equation has solution, the solution will be integer pair x and y which fulfills equation ax + by = c . Differently with Non-linear Diophantine equation, there is no standard method to find the solution. There are 3 possibilities related to the solution of Diophantine equation, either linear or non-linear. The solution may be single solution, multiple solutions or no solution. This research will discuss the solution of non-linear exponential Diophantine equation 13 x + 31 y = z 2 using the congruency theory. The methods used may be simulation, literature study and journal. Using the congruency theory, it is found that Non-Linear Exponential Diophantine equation 13 x + 31 y = z 2 has no solution, for x, y, z of integers.