Quadratic Diophantine equation in three variables How would one determine solutions to the following quadratic Diophantine equation in three variables:
$$x^2 + n^2y^2 \pm n^2y = z^2$$
where n is a known integer and $x$, $y$, and $z$ are unknown positive integers to be solved.
Ideally there would be a parametric solution for $x$, $y$, and $z$.
[Note that the expression $y^2 + y$ must be an integer from the series {2, 6, 12, 20, 30, 42 ...} and so can be written as either $y^2 + y$ or $y^2 - y$ (e.g., 12 = $3^2 + 3$ and 12 = $4^2 - 4$).  So I have written this as +/- in the equation above.]
Thanks,

This is a good problem as it requires to write the (Diophantine) parametric representations of the variables.
This method can be extended to n variables; one way of stating this result is to say there is an algorithm to construct a n variable  quadratic system  and if our equation fits this construction then the solutions are read off trivial.
I will explain this later.
