Bhaskara's lemma

Bhaskara's Lemma is an identity used as a lemma during the chakravala method. It states that:

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

for integers ${\displaystyle m,\,x,\,y,\,N,}$ and non-zero integer ${\displaystyle k}$.