Bhaskara's lemma explained

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( (mx + y)/(k) \right) ^{2} + (m ^{2} - N)/(k) = \left( (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( (mx + y)/(k) \right) ^{2} + (m ^{2} - N)/(k) = \left( (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