Bhaskara's lemma

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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.


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.)


  • 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).

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