I have been reading about this quadratic Diophantine equation of the form
$x^2 + axy + y^2 = z^2$
where x, y, z are integers to be solved and a is a given integer.
All integral solutions are given by
$x = k(an^2 - 2mn), y = k(m^2 - n^2), z = k(amn - m^2 - n^2)$ and
$x = k(m^2 - n^2), y = k(an^2 - 2mn), z = k(amn - m^2 - n^2)$
(due to diagonal symmetry in x and y)
where $m,n$ are integers with $\gcd(m,n) = 1,$ but $k \in \mathbb Q$ is rational such that $(a^2 - 4) \, k \in \mathbb Z.$ This is Theorem 2.3.2. on page 90 of An Introduction to Diophantine Equations by Andreescu, Andrica, and Cucurezeranu. (2010). EDIT BY WILL JAGY.
I have no problem understanding how the solution forms were derived; they were just basic algebraic manipulation. But then when it comes to the solutions in positive integers, the form becomes
$x = k(an^2 + 2mn), y = k(m^2 - n^2), z = k|amn + m^2 + n^2|$ and
$x = k(m^2 - n^2), y = k(an^2 + 2mn), z = k|amn + m^2 + n^2|$
where k, m, n are positive integers, an + 2 m > 0 and m > n.
What I can understand is that we apply modulus to the x, y, and z in the previous form to get the latter form (we want x, y, and z to be in positive integers), but I can't seem to understand how an + 2 m > 0 and m > n work to prove
$|x| = |k(an^2 - 2mn)| = kn|an - 2m| = kn(an + 2m) = k(an^2 + 2mn)$ and
$|z| = |k(amn - m^2 - n^2)| = k|amn + m^2 + n^2|$.
Can anyone help me on this? I've been pondering for almost a week. It's driving me crazy. Thank you in advance.