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Euler four-square identity (Theorem)

The Euler four-square identity simply states that

$ (x_1^2 + x_2^2 + x_3^2 + x_4^2)(y_1^2 + y_2^2 + y_3^2 + y_4^2) =$ $ (x_1y_1 + x_2y_2 + x_3y_3 + x_4y_4)^2 + (x_1y_2 - x_2y_1 + x_3y_4 - x_4y_3)^2$ $ + (x_1y_3 - x_3y_1 + x_4y_2 - x_2y_4)^2 + (x_1y_4 - x_4y_1 + x_2y_3 - x_3y_2)^2$

It may be derived from the property of quaternions that the norm of the product is equal to the product of the norms.



"Euler four-square identity" is owned by vitriol.
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See Also: Hamiltonian quaternions, Lagrange's four-square theorem, sums of two squares


Attachments:
proof of Euler four-square identity (Proof) by Thomas Heye
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Cross-references: product, norm, quaternions, property
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This is version 3 of Euler four-square identity, born on 2002-04-16, modified 2003-08-22.
Object id is 2839, canonical name is EulerFourSquareIdentity.
Accessed 4621 times total.

Classification:
AMS MSC: 11N32 (Number theory :: Multiplicative number theory :: Primes represented by polynomials; other multiplicative structure of polynomial values)
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