PlanetMath: sums of two squares

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sums of two squares

(Theorem)

Theorem 1 The set of the sums of two squares of integers is closed under multiplication; in fact we have the identical equation

$\displaystyle (a^2+b^2)(c^2+d^2) = (ac-bd)^2+(ad+bc)^2.$

This was presented by Leonardo Fibonacci in 1225 (in “Liber quadratorum”), but the original inventor was Brahmagupta.

The proof of the equation may utilize imaginary numbers as follows:

$\displaystyle (a^2+b^2)(c^2+d^2)$	$\displaystyle = (a+ib)(a-ib)(c+id)(c-id)$
	$\displaystyle = (a+ib)(c+id)(a-ib)(c-id)$
	$\displaystyle = [(ac-bd)+i(ad+bc)][(ac-bd)-i(ad+bc)]$
	$\displaystyle = (ac-bd)^2+(ad+bc)^2$

Note. The equation is the special case of Lagrange's identity.

"sums of two squares" is owned by pahio.

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Other names:

Brahmagupta's identity, Fibonacci's identity

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Cross-references: Lagrange's identity, imaginary numbers, proof, Fibonacci, equation, multiplication, closed under, integers
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This is version 23 of sums of two squares, born on 2004-05-10, modified 2008-05-18.
Object id is 5842, canonical name is SumsOfTwoSquares.
Accessed 3977 times total.

Classification:

AMS MSC:	11A67 (Number theory :: Elementary number theory :: Other representations)
	11E25 (Number theory :: Forms and linear algebraic groups :: Sums of squares and representations by other particular quadratic forms)

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