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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>
for integers <math>m,\, x,\, y,\, N,</math> and non-zero integer <math>k</math>.
 
==Proof==
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>.
 
:<math>\, Nx^2 + k = y^2\implies Nm^2x^2-N^2x^2+k(m^2-N) = m^2y^2-Ny^2</math>
:<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>
:<math>\implies \,N\left(\frac{mx + y}{k}\right)^2 + \frac{m^2 - N}{k} = \left(\frac{my + Nx}{k}\right)^2.</math>
 
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.)
 
==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==
*[http://www-groups.dcs.st-and.ac.uk/~history/Miscellaneous/Pearce/Lectures/Ch8_6.html Introduction to chakravala]
 
{{number-theoretic algorithms}}
 
{{DEFAULTSORT:Bhaskara's lemma, proof of}}
[[Category:Diophantine equations]]
[[Category:Number theoretic algorithms]]
[[Category:Lemmas]]
[[Category:Indian mathematics]]
[[Category:Articles containing proofs]]

Latest revision as of 08:36, 17 December 2014

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