Brahmagupta's identity

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In mathematics, Brahmagupta'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. Specifically:

<math>\left(a^2 + b^2\right)\left(c^2 + d^2\right) = \left(ac-bd\right)^2 + \left(ad+bc\right)^2.</math>

The identity holds in any commutative ring, but most usefully in the integers.

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.

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.

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