Bhaskara's lemma

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

$N{x}^2+k=y^2⟹N{\left(\frac{mx+y}{k}\right)}^2+\frac{m^2-N}{k}={\left(\frac{my+Nx}{k}\right)}^2$

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 $N^2{x}^2+2Nmxy+N{y}^2$, factor, and divide by $k^2$.

$N{x}^2+k=y^2⟹N{m}^2{x}^2-N^2{x}^2+k(m^2-N)=m^2{y}^2-N{y}^2$

$⟹N{m}^2{x}^2+2Nmxy+N{y}^2+k(m^2-N)=m^2{y}^2+2Nmxy+N^2{x}^2$

$⟹N(mx+y{)}^2+k(m^2-N)=(my+Nx{)}^2$

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

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