- Brahmagupta–Fibonacci identity
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.
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".
integercase this identity finds applications in number theoryfor 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.
The identity was discovered by
Brahmagupta(598–668), an Indian mathematician and astronomer. His " Brahmasphutasiddhanta" was translated from Sanskritinto Arabic by Mohammad al-Fazari, which was subsequently translated into Latinin 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.
Euler's four-square identityis 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 numbersnamely that:
by squaring both sides
and by the definition of absolute value,
Interpretation via norms
Therefore the identity is saying that
List of Indian mathematicians
* [http://planetmath.org/encyclopedia/BrahmaguptasIdentity.html Brahmagupta's identity at
* [http://www.pballew.net/fiboiden.html Brahmagupta-Fibonacci identity]
* [http://mathworld.wolfram.com/BrahmaguptaIdentity.html Brahmagupta Identity] on
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