# Brahmagupta-Fibonacci two-square identity

From Number

## Contents

## Statement

### As an identity

For all in a commutative ring:

### For products two at a time

In any commutative unital ring, if and can each be written as a sum of two squares, so can .

More concretely, if and , then .

### For products more than two at a time

In any commutative unital ring, if are elements each of which can be written as a sum of two squares, so can the product .

Note that this follows by induction from the statement for products two at a time.

## Interpretations

### In terms of complex numbers

This identity is motivated by complex numbers. In fact, it is precisely the statement that the modulus-squared operation for complex numbers is multiplicative. To see this note that:

- is the modulus-squared of the complex number
- is the modulus-squared of the complex number .
- is the modulus-squared of the complex number .

### In terms of trigonometry

This is based on the interpretation in terms of complex numbers, and relates the formula to the angle sum formulas for sine and cosine. *Fill this in later*