Bhaskara's lemma

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

\,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 $m,\,x,\,y,\,N,$ and non-zero integer $k$ .

Proof

The proof follows from simple algebraic manipulations as follows: multiply both sides of the equation by $m^{2}-N$ , add $N^{2}x^{2}+2Nmxy+Ny^{2}$ , factor, and divide by $k^2$ .

\,Nx^{2}+k=y^{2}\implies Nm^{2}x^{2}-N^{2}x^{2}+k(m^{2}-N)=m^{2}y^{2}-Ny^{2}

\implies Nm^{2}x^{2}+2Nmxy+Ny^{2}+k(m^{2}-N)=m^{2}y^{2}+2Nmxy+N^{2}x^{2}

\implies N(mx+y)^{2}+k(m^{2}-N)=(my+Nx)^{2}

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

So long as neither $k$ nor $m^{2}-N$ are zero, the implication goes in both directions. (Note also that the lemma holds for real or complex numbers as well as integers.)

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

Introduction to chakravala

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