Brahmagupta's identity

In mathematics, the Brahmagupta identity states that the product of two numbers, each of which is the sum of two squares, is also the sum of two squares. Specifically:

(a2+b2)(c2+d2)=(ac− − bd)2+(ad+bc)2(1)=(ac+bd)2+(ad− − bc)2.(2){displaystyle {begin{aligned}left(a^{2}+b^{2}right)left(c^{2}+d^{2}{2}{2}{2}{right}{left(ac-bdright)^{2}{2}{2}{qquad qquad qquad={2}{

The identity is true in any commutative ring, but has its greatest utility in the ring of integers.

The identity was named after the Indian mathematician and astronomer Brahmagupta (598-668).

See also Euler's four-square identity. There is a similar eight-square identity that follows for octonions, but it is not especially interesting for integers because every positive integer is a sum of four squares.