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 (inLiber quadratorum), but was known also by Brahmaguptaand already by Diophantus of Alexandria (III book of hisArithmetica).

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 ofsquares.  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 $8^2+1^2$.