Bhaskara's Lemma

''Bhaskara's'' Lemma is an identity used as a lemma during the chakravala method. It states that:
:$\, Nx^2 + k = y^2\implies \,N\left(\frac\right)^2 + \frac = \left(\frac\right)^2$
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 $k^2$.

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

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

Diophantine equations
Number theoretic algorithms
Lemmas in algebra
Indian mathematics
Articles containing proofs

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