Brahmagupta–Fibonacci identity

In algebra, Brahmagupta's identity, also sometimes called Fibonacci's identity, implies that the product of two sums 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. The identity is a special case ("n" = 2) of Lagrange's identity.

Specifically:

:

For example,

:$(1^2\; +\; 4^2)(2^2\; +\; 7^2)\; =\; 30^2\; +\; 1^2\; =\; 26^2\; +\; 15^2.,$

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 4"n" + 1 is also a sum of two squares.

History

The identity was discovered by Brahmagupta (598–668), an Indian mathematician and astronomer. His "Brahmasphutasiddhanta" was translated from Sanskrit into Arabic by Mohammad al-Fazari, which was subsequently translated into Latin in 1126. [George G. Joseph (2000). "The Crest of the Peacock", p. 306. Princeton University Press. ISBN 0691006598.] 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:

:$|\; a+bi\; |\; |\; c+di\; |\; =\; |\; (a+bi)(c+di)\; |\; ,$

since

:$|\; a+bi\; |\; |\; c+di\; |\; =\; |\; (ac-bd)+i(ad+bc)\; |,,$

by squaring both sides

:$|\; a+bi\; |^2\; |\; c+di\; |^2\; =\; |\; (ac-bd)+i(ad+bc)\; |^2,,$

and by the definition of absolute value,

:$(a^2+b^2)(c^2+d^2)=\; (ac-bd)^2+(ad+bc)^2.\; ,$

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

: $N(a+bi)\; =\; a^2\; +\; b^2\; ext\{\; and\; \}N(c+di)\; =\; c^2\; +\; d^2,\; ,$

and also

: $N((a+bi)(c+di))\; =\; N((ac-bd)+i(ad+bc))\; =\; (ac-bd)^2\; +\; (ad+bc)^2.\; ,$

Therefore the identity is saying that

: $N((a+bi)(c+di))\; =\; N(a+bi)\; cdot\; N(c+di).\; ,$

ee also

* Brahmagupta matrix
* Indian mathematics
* List of Indian mathematicians

References

External links

* [http://planetmath.org/encyclopedia/BrahmaguptasIdentity.html Brahmagupta's identity at PlanetMath]
* [http://www.pballew.net/fiboiden.html Brahmagupta-Fibonacci identity]
* [http://mathworld.wolfram.com/BrahmaguptaIdentity.html Brahmagupta Identity] on MathWorld