sums of two squares


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

(a2+b2)(c2+d2)=(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 integersMathworldPlanetmath as follows:

(a2+b2)(c2+d2) =(a+ib)(a-ib)(c+id)(c-id)

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

(a2+b2)(c2+d2)=(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


attains the two forms 42+72 and 82+12.

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 TheoremMathworldPlanetmath
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