The Solution for the Non linear Diophantine Equation (7k -1)x +(7k)y = z2 with k as the positive even whole number

Abstract

This research may provide the solutions (if any) from the Non-linear Diophantine equation (7k — 1)x + (7k )y = Z2.There are 3 possibilities to determine the solutions from the Non-linear Diophantine equation: single solution, multiple solutions, and no solution. The research method is conducted in two stages: first, using a simulation to determine the solutions (if any) from the Non-linear Diophantine equation (7k — 1)x + (7k )y = z2; second, using Catalan conjecture and characteristics of congruency theory, which is proven that the Non-k linear Diophantine equation has a single solution for x, y, z as non-negative whole numbers and k as the positive even whole number.