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.
Both (1) and (2) can be verified by expanding each side of the equation. Also, (2) can be obtained from (1) by changing b to −b.
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 See also
- 6 References
- 7 External links
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. The identity later appeared in Fibonacci's Book of Squares in 1225.
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
by squaring both sides
and by the definition of absolute value,
Interpretation via norms
Therefore the identity is saying that
- ^ George G. Joseph (2000). The Crest of the Peacock, p. 306. Princeton University Press. ISBN 0691006598.
- Nrahmagupta's identity at PlanetMath
- Brahmagupta-Fibonacci identity
- Brahmagupta Identity on MathWorld