Brahmagupta's identity

In algebra, Brahmagupta's identity says that, for given $n$ , the product of two numbers of the form $a^{2}+nb^{2}$ is itself a number of that form. In other words, the set of such numbers is closed under multiplication. Specifically:

{\begin{aligned}\left(a^{2}+nb^{2}\right)\left(c^{2}+nd^{2}\right)&{}=\left(ac-nbd\right)^{2}+n\left(ad+bc\right)^{2}&&&(1)\\&{}=\left(ac+nbd\right)^{2}+n\left(ad-bc\right)^{2},&&&(2)\end{aligned}}

Both (1) and (2) can be verified by expanding each side of the equation. Also, (2) can be obtained from (1), or (1) from (2), by changing b to −b.

This identity holds in both the ring of integers and the ring of rational numbers, and more generally in any commutative ring.

History

The identity is a generalization of the so-called Fibonacci identity (where n=1) which is actually found in Diophantus' Arithmetica (III, 19). That identity was rediscovered by Brahmagupta (598–668), an Indian mathematician and astronomer, who generalized it and used it in his study of what is now called Pell's equation. His Brahmasphutasiddhanta was translated from Sanskrit into Arabic by Mohammad al-Fazari, and was subsequently translated into Latin in 1126.^[1] The identity later appeared in Fibonacci's Book of Squares in 1225.

Application to Pell's equation

In its original context, Brahmagupta applied his discovery to the solution of what was later called Pell's equation, namely x² − Ny² = 1. Using the identity in the form

(x_{1}^{2}-Ny_{1}^{2})(x_{2}^{2}-Ny_{2}^{2})=(x_{1}x_{2}+Ny_{1}y_{2})^{2}-N(x_{1}y_{2}+x_{2}y_{1})^{2},

he was able to "compose" triples (x₁, y₁, k₁) and (x₂, y₂, k₂) that were solutions of x² − Ny² = k, to generate the new triple

(x_{1}x_{2}+Ny_{1}y_{2}\,,\,x_{1}y_{2}+x_{2}y_{1}\,,\,k_{1}k_{2}).

Not only did this give a way to generate infinitely many solutions to x² − Ny² = 1 starting with one solution, but also, by dividing such a composition by k₁k₂, integer or "nearly integer" solutions could often be obtained. The general method for solving the Pell equation given by Bhaskara II in 1150, namely the chakravala (cyclic) method, was also based on this identity.^[2]

References

^ George G. Joseph (2000). The Crest of the Peacock, p. 306. Princeton University Press. ISBN 0-691-00659-8.
^ John Stillwell (2002), Mathematics and its history (2 ed.), Springer, pp. 72–76, ISBN 978-0-387-95336-6

External links

Brahmagupta's identity at PlanetMath
Brahmagupta Identity on MathWorld
A Collection of Algebraic Identities Archived 2012-03-06 at the Wayback Machine

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