Brahmagupta-Fibonacci Identity

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

Then:

This is an example of a more general identity:

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

Then:
 * $\displaystyle \prod_{j=1}^n \left({a_j^2 + b_j^2}\right) = c^2 + d^2$

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.

More generally:
 * $\displaystyle \prod_{j=1}^n \left({a_j^2 + n b_j^2}\right) = c^2 + n d^2$

where $c, d \in \Z$.

That is, the set of all numbers of the form $x^2 + n y^2$ is closed under multiplication.

Proof
Setting $b = -b$ in the above gives the identity:


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

The identities:

follow from the above by setting $n = 1$.

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.

Both of these described this identity in their writings:


 * 628: Brahmagupta: Brahmasphutasiddhanta (The Opening of the Universe)
 * 1225: Fibonacci: Liber quadratorum (The Book of Squares)