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,
Considering this as a quadratic in $y$, it is sufficient (and necessary) that 

$n^4 - 4n^2(x^2-z^2)$ is a perfect square.

i.e.

$n^2 - 4(x^2 - z^2)$ is a perfect square, say $q^2$.

Rewriting gives us

$(n-q)(n+q) = 4 (x^2 - z^2)$

If $n=2k$ is even, then $q$ needs to be even too (say $2m$) and

$(k-m)(k+m) = (x-z)(x+z)$

If $n=2k+1$ is odd, then $q$ needs to be odd too (say $2m+1$) and

$(k-m)(k+m+1) = (x-z)(x+z)$

Thus you can pick any $m$, and try to factor the left hand side above (choosing the right one depending on the parity of $n$) into two terms of the same parity.
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.