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| '''''Bhaskara's'' Lemma''' is an identity used as a lemma during the [[chakravala method]]. It states that:
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| :<math>\, 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</math>
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| for integers <math>m,\, x,\, y,\, N,</math> and non-zero integer <math>k</math>.
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| ==Proof==
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| The proof follows from simple algebraic manipulations as follows: multiply both sides of the equation by <math>m^2-N</math>, add <math>N^2x^2+2Nmxy+Ny^2</math>, factor, and divide by <math>k^2</math>.
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| :<math>\, Nx^2 + k = y^2\implies Nm^2x^2-N^2x^2+k(m^2-N) = m^2y^2-Ny^2</math>
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| :<math>\implies Nm^2x^2+2Nmxy+Ny^2+k(m^2-N) = m^2y^2+2Nmxy+N^2x^2</math> | |
| :<math>\implies N(mx+y)^2+k(m^2-N) = (my+Nx)^2</math>
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| :<math>\implies \,N\left(\frac{mx + y}{k}\right)^2 + \frac{m^2 - N}{k} = \left(\frac{my + Nx}{k}\right)^2.</math>
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| So long as neither <math>k</math> nor <math>m^2-N</math> are zero, the implication goes in both directions. (Note also that the lemma holds for real or complex numbers as well as integers.)
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| ==References==
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| *C. O. Selenius, "Rationale of the chakravala process of Jayadeva and Bhaskara II", ''Historia Mathematica'', 2 (1975), 167-184.
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| *C. O. Selenius, ''Kettenbruch theoretische Erklarung der zyklischen Methode zur Losung der Bhaskara-Pell-Gleichung'', Acta Acad. Abo. Math. Phys. 23 (10) (1963).
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| *George Gheverghese Joseph, ''The Crest of the Peacock: Non-European Roots of Mathematics'' (1975).
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| ==External links==
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| *[http://www-groups.dcs.st-and.ac.uk/~history/Miscellaneous/Pearce/Lectures/Ch8_6.html Introduction to chakravala]
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| {{number-theoretic algorithms}}
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| {{DEFAULTSORT:Bhaskara's lemma, proof of}}
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| [[Category:Diophantine equations]]
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| [[Category:Number theoretic algorithms]]
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| [[Category:Lemmas]] | |
| [[Category:Indian mathematics]]
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| [[Category:Articles containing proofs]]
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