Solutions to non-linear diophantine equation (5k-1)x + (5k)y = z2 with k is positive even integer

Abstract

This research seeks for a solution (if any) to non-linear Diophantine equation (5k-1)x + (5k)y = z2. There are 3 possibilities of solution to the non-linear Diophantine equation, which are single solution, many solutions, or no solution. The research methodology is conducted in two stages has t, which are using simulation to seek for a solution (if any) to non-linear Diophantine equation (5k-1)x + (5k)y = z2 and using Catalan’s conjecture and characteristics of congruence theory. It is proven that the non-linear Diophantine equation has single solution (x, y, z) = (1, 0, 5k/2) for non-negative integer x, y, z and positive even of equal to or higher than.