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
(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).
Note 1. The equation (1) is the special case
of Lagrange’s identity.
Note 2. Similarly as (1), one can derive the identity
(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 and .
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 |