Mathematical lemma
Bhaskara's Lemma is an identity used as a lemma during the chakravala method. It states that:
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c804a445c6839b266d56a715894f36d6808eabf9)
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
.
![{\displaystyle \,Nx^{2}+k=y^{2}\implies Nm^{2}x^{2}-N^{2}x^{2}+k(m^{2}-N)=m^{2}y^{2}-Ny^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44c6006241e4f8f6102b22028a29d3c9aba0b344)
![{\displaystyle \implies Nm^{2}x^{2}+2Nmxy+Ny^{2}+k(m^{2}-N)=m^{2}y^{2}+2Nmxy+N^{2}x^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e4a364862c5aa7bb23720b365410333025ae73d)
![{\displaystyle \implies N(mx+y)^{2}+k(m^{2}-N)=(my+Nx)^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67f37eaa4d477321107ca275c9df636a664be176)
![{\displaystyle \implies \,N\left({\frac {mx+y}{k}}\right)^{2}+{\frac {m^{2}-N}{k}}=\left({\frac {my+Nx}{k}}\right)^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a40d9db0818dbf11f9f030bc6af097ba8d8cbc5f)
So long as neither
nor
are zero, the implication goes in both directions. (The lemma holds for real or complex numbers as well as integers.)
References[edit]
- 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[edit]