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. (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