Brahmagupta-Fibonacci Identity

Theorem
Let $$a, b, c, d$$ be numbers.

Then $$\left({a^2 + b^2}\right) \left({c^2 + d^2}\right) = \left({a c + b d}\right)^2 + \left({a d - b c}\right)^2$$.

Corollary
Let $$a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$$ be integers.

Then $$\prod_{j=1}^n \left({a_j^2 + b_j^2}\right) = c^2 + d^2$$, where $$c, d \in \Z$$.

What this says is that the product of any number of sums of two squares is also a sum of two squares.

Proof
$$ $$ $$

Proof of Corollary
Follows by induction from the main result.

Note
This identity is also known as Fibonacci's Identity, and is a special case for $$n = 2$$ of Lagrange's Identity.

Named for Brahmagupta‎ and Fibonacci‎, both of whom described this identity in their writings.