Editing Brahmagupta–Fibonacci identity (section)

{{short description|Expression of a product of sums of squares as a sum of squares}}
In [[algebra]], the '''Brahmagupta–Fibonacci identity'''<ref>{{Cite web|url=http://www.cut-the-knot.org/m/Algebra/BrahmaguptaFibonacci.shtml|title = Brahmagupta-Fibonacci Identity}}</ref><ref>Marc Chamberland: ''Single Digits: In Praise of Small Numbers''. Princeton University Press, 2015, {{ISBN|9781400865697}}, p. 60</ref> expresses the product of two sums of two squares as a sum of two squares in two different ways. Hence the set of all sums of two squares is [[closure (mathematics)|closed]] under multiplication. Specifically, the identity says
:<math>\begin{align}
\left(a^2 + b^2\right)\left(c^2 + d^2\right) & {}= \left(ac-bd\right)^2 + \left(ad+bc\right)^2 & & (1) \\
                                             & {}= \left(ac+bd\right)^2 + \left(ad-bc\right)^2. & & (2)
\end{align}</math>
For example,
:<math>(1^2 + 4^2)(2^2 + 7^2) = 26^2 + 15^2 = 30^2 + 1^2.</math>

The identity is also known as the '''Diophantus identity''',<ref name=stillwell2>{{Harvnb|Stillwell|2002|p =76}}</ref><ref>[[Daniel Shanks]], Solved and unsolved problems in number theory, p.209, American Mathematical Society, Fourth edition 1993.</ref>  as it was first proved by [[Diophantus|Diophantus of Alexandria]].  It is a special case of [[Euler's four-square identity]], and also of [[Lagrange's identity]]. 

[[Brahmagupta]] proved and used a more general [[Brahmagupta identity]], stating
:<math>\begin{align}
\left(a^2 + nb^2\right)\left(c^2 + nd^2\right) & {}= \left(ac-nbd\right)^2 + n\left(ad+bc\right)^2 & & (3) \\
                                               & {}= \left(ac+nbd\right)^2 + n\left(ad-bc\right)^2. & & (4)
\end{align}</math>
This shows that, for any fixed ''A'', the set of all numbers of the form ''x''<sup>2</sup>&nbsp;+&nbsp;''Ay''<sup>2</sup> is closed under multiplication.

These identities hold for all [[integers]], as well as all [[rational number]]s; more generally, they are true in any [[commutative ring]].  All four forms of the identity can be verified by [[polynomial expansion|expanding]] each side of the equation.  Also, (2) can be obtained from (1), or (1) from (2), by changing ''b'' to&nbsp;&minus;''b'', and likewise with (3) and (4).