Bhaskara's lemma

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

for integers and non-zero integer .


The proof follows from simple algebraic manipulations as follows: multiply both sides of the equation by , add , factor, and divide by .

So long as neither nor 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).

External links

Template:Number-theoretic algorithms