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 $N^2x^2+2Nmxy+Ny^2$, factor, and divide by $k^2$.

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

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).