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

This research may provide the solutions (if any) from the Non-linear Diophantine equation (7 k — 1) x + (7 k ) y = Z 2 .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 (7 k — 1) x + (7 k ) y = z 2 ; second, using Catalan conjecture and characteristics of congruency theory, which is proven that the Non- k linear Diophantine equation has a single solution ( x , y , z ) = ( 1 , 0 , 7 k 2 ) for x , y , z as non-negative whole numbers and k as the positive even whole number.