Bhaskara's lemma

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

[math]\displaystyle{ \, 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]\displaystyle{ m,\, x,\, y,\, N, }[/math] and non-zero integer [math]\displaystyle{ k }[/math].

Proof

The proof follows from simple algebraic manipulations as follows: multiply both sides of the equation by [math]\displaystyle{ m^2-N }[/math], add [math]\displaystyle{ N^2x^2+2Nmxy+Ny^2 }[/math], factor, and divide by [math]\displaystyle{ k^2 }[/math].

[math]\displaystyle{ \, Nx^2 + k = y^2\implies Nm^2x^2-N^2x^2+k(m^2-N) = m^2y^2-Ny^2 }[/math]

[math]\displaystyle{ \implies Nm^2x^2+2Nmxy+Ny^2+k(m^2-N) = m^2y^2+2Nmxy+N^2x^2 }[/math]

[math]\displaystyle{ \implies N(mx+y)^2+k(m^2-N) = (my+Nx)^2 }[/math]

[math]\displaystyle{ \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]\displaystyle{ k }[/math] nor [math]\displaystyle{ m^2-N }[/math] are zero, the implication goes in both directions. (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

• Introduction to chakravala