Bhaskara's Lemma

Lemma
As long as $$k \ne 0$$:


 * $$N x^2 + k = y^2 \implies N \left({\frac{mx + y}{k}}\right)^2 + \frac{m^2 - N}{k} = \left({\frac{my + Nx}{k}}\right)^2$$

for any integer $$m$$.

Proof
$$ $$ $$ $$ $$

The implication goes the other way if $$m^2 - N \ne 0$$.