Euler's four-square identity
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In mathematics, Euler's four-square 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:
- <math>(a_1^2+a_2^2+a_3^2+a_4^2)(b_1^2+b_2^2+b_3^2+b_4^2)\,<math>
- <math>=(a_1 b_1-a_2 b_2 - a_3 b_3 - a_4 b_4)^2 + (a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3)^2\,<math>
- <math>+\,(a_1 b_3 - a_2 b_4 + a_3 b_1 + a_4 b_2)^2 + (a_1 b_4 + a_2 b_3 - a_3 b_2 + a_4 b_1)^2\,<math>
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.es:Indentidad de los cuatro colores de Euler it:Identità dei quattro quadrati di Eulero sl:Eulerjeva enakost štirih kvadratov