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'''''Bhaskara's'' Lemma''' is an identity used as a lemma during the chakravala method It states that:
$Nx^2 + k = y^2implies Nleft\left(frac\left\{mx + y\right\}\left\{k\right\}right\right)^2 + frac\left\{m^2 - N\right\}\left\{k\right\} = left\left(frac\left\{my + Nx\right\}\left\{k\right\}right\right)^2$

## Proof

We begin with an identity verified by expansion (or substitution into the Brahmagupta-Fibonacci identity with $a=mc=yb=isqrt\left\{N\right\}c=ixsqrt\left\{N\right\}$) :
$N\left(mx+y\right)^2-\left(my+Nx\right)^2=\left(m^2-N\right)\left(y^2-Nx^2\right)Longleftrightarrow frac\left\{N\left(mx+y\right)^2-\left(my+Nx\right)^2\right\}\left\{y^2-Nx^2\right\}=m^2-N$
Since $y^2-Nx^2=k$ this yields:
$frac\left\{N\left(mx+y\right)^2-\left(my+Nx\right)^2\right\}\left\{k\right\}=m^2-NLongleftrightarrowfrac\left\{N\left(mx+y\right)^2-\left(my+Nx\right)^2\right\}\left\{k^2\right\}=frac\left\{m^2-N\right\}\left\{k\right\}$
After rearranging and factoring the denominators this yields Bhaskara's Lemma:
$Nleft\left(frac\left\{mx + y\right\}\left\{k\right\}right\right)^2 + frac\left\{m^2 - N\right\}\left\{k\right\} = left\left(frac\left\{my + Nx\right\}\left\{k\right\}right\right)^2$

## References

• C O. Selenius "Rationale of the chakravala process of Jayadeva and Bhaskara II" Historia Mathematica 2 (1975) 167-184
• C O. Selenius Kettenbruch theoretische Erklarung der zyklischen Methode zur Losung der Bhaskara-Pell-Gleichung Acta Acad Abo Math Phys 23 (10) (1963)
• George Gheverghese Joseph The Crest of the Peacock: Non-European Roots of Mathematics (1975)