Solutions to Non-Linear Diophantine Equation Pχ+ (P+5)y =z2 with P is Mersenne Prime

This research seeks for a solution (if any) to non-linear Diophantine equation 2 ( 5) x y p p z with p is Mersenne prime. 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, which are using simulation to seek for solution (if any) to non-linear Diophantine equation 2 ( 5) x y p p z with p is Mersenne prime and using Catalan’s conjecture and characteristics of congruency theory. it is proven that the non-linear Diophantine equation has no solution for p 3.