Existence and Number of Solutions of Diophantine Quadratic Equations with Two Unknowns in $Z$ and $N$

In this short note we study the existence and number of solutions in the set of integers ($Z$) and in the set of natural numbers ($N$) of Diopahntine Equations of second degree with two variables of the general form $a{x}^2-b{y}^2=c$.