Brahmagupta–Fibonacci identity

In algebra, Brahmagupta's identity, also called Fibonacci's identity, implies that the product of two sums each of two squares is itself a sum of two squares. In other words, the set of all sums of two squares is closed under multiplication. Specifically:

$\begin{align} \left(a^2 %2B b^2\right)\left(c^2 %2B d^2\right) & {}= \left(ac-bd\right)^2 %2B \left(ad%2Bbc\right)^2 \ \qquad\qquad(1) \\ & {}= \left(ac%2Bbd\right)^2 %2B \left(ad-bc\right)^2.\qquad\qquad(2) \end{align}$

For example,

$(1^2 %2B 4^2)(2^2 %2B 7^2) = 26^2 %2B 15^2 = 30^2 %2B 1^2.\,$

The identity is a special case (n = 1) of Lagrange's identity, and is first found in Diophantus. Brahmagupta proved and used a more general identity, equivalent to

$\begin{align} \left(a^2 %2B nb^2\right)\left(c^2 %2B nd^2\right) & {}= \left(ac-nbd\right)^2 %2B n\left(ad%2Bbc\right)^2 \ \qquad\qquad(3) \\ & {}= \left(ac%2Bnbd\right)^2 %2B n\left(ad-bc\right)^2,\qquad\qquad(4) \end{align}$

showing that the set of all numbers of the form $x^2 %2B ny^2$ is closed under multiplication.

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.

In the integer case this identity finds applications in number theory for example when used in conjunction with one of Fermat's theorems it proves that the product of a square and any number of primes of the form 4n + 1 is also a sum of two squares.

1 History
2 Related identities
3 Relation to complex numbers
4 Interpretation via norms
5 Application to Pell's equation
6 See also
7 References
8 External links

History

The identity is first found in Diophantus's Arithmetica (III, 19). The 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 erroneously 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.

Related identities

Euler's four-square identity is an analogous identity involving four squares instead of two that is related to quaternions. There is a similar eight-square identity derived from the Cayley numbers which has connections to Bott periodicity.

Relation to complex numbers

If a, b, c, and d are real numbers, this identity is equivalent to the multiplication property for absolute values of complex numbers namely that:

$| a%2Bbi | | c%2Bdi | = | (a%2Bbi)(c%2Bdi) | \,$

since

$| a%2Bbi | | c%2Bdi | = | (ac-bd)%2Bi(ad%2Bbc) |,\,$

by squaring both sides

$| a%2Bbi |^2 | c%2Bdi |^2 = | (ac-bd)%2Bi(ad%2Bbc) |^2,\,$

and by the definition of absolute value,

$(a^2%2Bb^2)(c^2%2Bd^2)= (ac-bd)^2%2B(ad%2Bbc)^2. \,$

Interpretation via norms

In the case that the variables a, b, c, and d are rational numbers, the identity may be interpreted as the statement that the norm in the field Q(i) is multiplicative. That is, we have

$N(a%2Bbi) = a^2 %2B b^2 \text{ and }N(c%2Bdi) = c^2 %2B d^2, \,$

and also

$N((a%2Bbi)(c%2Bdi)) = N((ac-bd)%2Bi(ad%2Bbc)) = (ac-bd)^2 %2B (ad%2Bbc)^2. \,$

Therefore the identity is saying that

$N((a%2Bbi)(c%2Bdi)) = N(a%2Bbi) \cdot N(c%2Bdi). \,$

Application to Pell's equation

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

$(x_1^2 - Ny_1^2)(x_2^2 - Ny_2^2) = (x_1x_2 %2B Ny_1y_2)^2 - N(x_1y_2 %2B x_2y_1)^2, \,$

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

$(x_1x_2 %2B Ny_1y_2 \,,\, x_1y_2 %2B x_2y_1 \,,\, k_1k_2).$

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 0691006598.
^ John Stillwell (2002), Mathematics and its history (2 ed.), Springer, pp. 72–76, ISBN 9780387953366, http://books.google.com/books?id=WNjRrqTm62QC&pg=PA72