Euler's foursquare identity
In mathematics, Euler's foursquare identity says that the product of two numbers, each of which being a sum of four squares, is itself a sum of four squares. Specifically:


Euler wrote about this identity in
1750. It can be proven with
elementary algebra and holds in every
commutative ring. If the
as and
bs are
real numbers, a more elegant proof is available: the identity expresses the fact that the absolute value of the product of two
quaternions is equal to the product of their absolute values, in the same way that
Brahmagupta's identity does for
complex numbers.
The identity was used by Lagrange to prove his four square theorem.