Brahmagupta-Fibonacci identity
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In algebra, Brahmagupta's identity, also sometimes called Fibonacci's identity, says that the product of two numbers, each of which is a sum of two squares, is itself a sum of two squares (and in two different ways). In other words, the set of all sums of two squares is closed under multiplication.
Specifically:
For example,
The identity is used in proofs of Fermat's theorem on sums of two squares.
The identity holds in any commutative ring, but most usefully in the integers.
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[edit] History
The identity was discovered by Brahmagupta (598-668), an Indian mathematician and astronomer. It was later translated to Arabic and Persian, and then translated to Latin by Leonardo of Pisa (1170-1250) also known as Fibonacci. It appeared in Fibonacci's Book of Squares in 1225. It may also have been known to Diophantus in the 3rd century.
[edit] Related identities
Euler's four-square identity is an analogous identity involving four squares instead of two. There is a similar eight-square identity derived from the Cayley numbers which has connections to Bott periodicity.
[edit] 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) = a2 + b2
and
- N(c + di) = c2 + d2,
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
