Proof of Bhaskara's lemma

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

Proof

With simple manipulations on both sides of the equation of multiplying by $m^2-N$, adding $N^2x^2+2Nmxy+Ny^2$, factorising and then dividing by nonzero $k^2$:

:$,\; Nx^2\; +\; k\; =\; y^2implies\; Nm^2x^2-N^2x^2+k(m^2-N)\; =\; m^2y^2-Ny^2$:$implies\; Nm^2x^2+2Nmxy+Ny^2+k(m^2-N)\; =\; m^2y^2+2Nmxy+N^2x^2$ :$implies\; N(mx+y)^2+k(m^2-N)\; =\; (my+Nx)^2$ :$implies\; ,Nleft(frac\{mx\; +\; y\}\{k\}\; ight)^2\; +\; frac\{m^2\; -\; N\}\{k\}\; =\; left(frac\{my\; +\; Nx\}\{k\}\; ight)^2.$

So long as neither $k$ nor $m^2-N$ are zero, the implication goes in both directions

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).

External links

* [http://www-groups.dcs.st-and.ac.uk/~history/Miscellaneous/Pearce/Lectures/Ch8_6.html Introduction to chakravala]