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Bhaskara's lemma

    '''''Bhaskara's'' Lemma''' is an identity used as a lemma during the chakravala method It states that:
    Nx^2 + k = y^2implies Nleft(frac{mx + y}{k}right)^2 + frac{m^2 - N}{k} = left(frac{my + Nx}{k}right)^2

    Proof

    We begin with an identity verified by expansion (or substitution into the Brahmagupta-Fibonacci identity with a=mc=yb=isqrt{N}c=ixsqrt{N}) :
    N(mx+y)^2-(my+Nx)^2=(m^2-N)(y^2-Nx^2)Longleftrightarrow frac{N(mx+y)^2-(my+Nx)^2}{y^2-Nx^2}=m^2-N
    Since y^2-Nx^2=k this yields:
    frac{N(mx+y)^2-(my+Nx)^2}{k}=m^2-NLongleftrightarrowfrac{N(mx+y)^2-(my+Nx)^2}{k^2}=frac{m^2-N}{k}
    After rearranging and factoring the denominators this yields Bhaskara's Lemma:
    Nleft(frac{mx + y}{k}right)^2 + frac{m^2 - N}{k} = left(frac{my + Nx}{k}right)^2

    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)

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