sums of two squares

Theorem.

The set of the sums of two squares of integers is closed under multiplication; in fact we have the identical equation

$(a^2+b^2)(c^2+d^2)={(ac-bd)}^2+{(ad+bc)}^2.$ (1)

This was presented by Leonardo Fibonacci in 1225 (in Liber quadratorum), but was known also by Brahmagupta and already by Diophantus of Alexandria (III book of his Arithmetica).

The proof of the equation may utilize Gaussian integers as follows:

$(a^2+b^2)(c^2+d^2)$ $\mathrm{\hspace{0.33em}}=(a+ib)(a-ib)(c+id)(c-id)$

$\mathrm{\hspace{0.33em}}=(a+ib)(c+id)(a-ib)(c-id)$

$\mathrm{\hspace{0.33em}}=[(ac-bd)+i(ad+bc)][(ac-bd)-i(ad+bc)]$

$\mathrm{\hspace{0.33em}}={(ac-bd)}^2+{(ad+bc)}^2$

Note 1.  The equation (1) is the special case  $n=2$  of Lagrange’s identity.

Note 2.  Similarly as (1), one can derive the identity

$(a^2+b^2)(c^2+d^2)={(ac+bd)}^2+{(ad-bc)}^2.$ (2)

Thus in most cases, we can get two different nontrivial sum forms (i.e. without a zero addend) for a given product of two sums of squares.  For example, the product

$$65=5\cdot 13=(2^2+1^2)(3^2+2^2)$$

attains the two forms $4^2+7^2$ and $8^2+1^2$.

Title	sums of two squares
Canonical name	SumsOfTwoSquares
Date of creation	2013-11-19 16:28:21
Last modified on	2013-11-19 16:28:21
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	33
Author	pahio (2872)
Entry type	Theorem
Classification	msc 11A67
Classification	msc 11E25
Synonym	Diophantus’ identity
Synonym	Brahmagupta’s identity
Synonym	Fibonacci’s identity
Related topic	EulerFourSquareIdentity
Related topic	TheoremsOnSumsOfSquares
Related topic	DifferenceOfSquares