Talk:Bhaskara's lemma

This is my first page addition on wikipedia. I copied the references from the "parent" page this came from. Is that okay or should references be cited only if they were used to research the article? 05:59, 29 September 2007 User:800km3rk

References should help later readers verify statements in the article. So they need to be relevant to this article. --Rumping (talk) 13:57, 14 August 2008 (UTC)

First line of proof

I got lost in the first line of the proof. Where did

$N(mx+y)^{2}-(my+Nx)^{2}=-(m^{2}-N)(y^{2}-Nx^{2})$

come from? I have provided what I think is a simpler alternative proof, and if there are no objections I will delete the original later. --Rumping (talk) 13:57, 14 August 2008 (UTC)

There has been no objection in six weeks so I have removed the original. It follows here in case anybody want to see it. --Rumping (talk) 22:36, 29 September 2008 (UTC)

Earlier version of proof

We begin with an identity, verified by expansion (or substitution into the Brahmagupta-Fibonacci identity with $a=m,c=y,b=i{\sqrt {N}},c=ix{\sqrt {N}}$) :

$N(mx+y)^{2}-(my+Nx)^{2}=-(m^{2}-N)(y^{2}-Nx^{2})\Longleftrightarrow {\frac {N(mx+y)^{2}-(my+Nx)^{2}}{y^{2}-Nx^{2}}}=-(m^{2}-N).$

Since $y^{2}-Nx^{2}=k$, we have that:

${\frac {N(mx+y)^{2}-(my+Nx)^{2}}{k}}=-(m^{2}-N)\Longleftrightarrow {\frac {N(mx+y)^{2}-(my+Nx)^{2}}{k^{2}}}={\frac {-(m^{2}-N)}{k}}.$

Suitable re-arrangement of this equation yields Bhaskara's Lemma:

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