Editing Brahmagupta–Fibonacci identity

{{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).

==History==
The identity first appeared in [[Diophantus]]' ''[[Arithmetica]]'' (III, 19), of the third century A.D.
It was rediscovered by Brahmagupta (598&ndash;668), an [[Indian mathematicians|Indian mathematician]] and [[Indian astronomy|astronomer]], who generalized it to [[Brahmagupta's identity]], and used it in his study of what is now called [[Pell's equation]]. His ''[[Brahmasphutasiddhanta]]'' was translated from [[Sanskrit]] into [[Arabic language|Arabic]] by [[Mohammad al-Fazari]], and was subsequently translated into [[Latin]] in 1126.<ref name=Joseph>{{Harvnb|Joseph|2000|p=306}}</ref> The identity was introduced in western Europe in 1225 by [[Fibonacci]], in ''[[The Book of Squares]]'', and, therefore, the identity has been often attributed to him.

==Related identities==

Analogous identities are [[Euler's four-square identity|Euler's four-square]] related to [[quaternions]], and [[Degen's eight-square identity|Degen's eight-square]] derived from the [[octonions]] which has connections to [[Bott periodicity]]. There is also [[Pfister's sixteen-square identity]], though it is no longer bilinear.

These identities are strongly related with [[Hurwitz's theorem (composition algebras)|Hurwitz's classification]] of [[composition algebra]]s.

The Brahmagupta–Fibonacci identity is a special form of [[Lagrange's identity]], which is itself a special form of [[Binet–Cauchy identity]], in turn a special form of the [[Cauchy–Binet formula]] for matrix determinants.

== Multiplication of complex numbers ==

If ''a'', ''b'', ''c'', and ''d'' are [[real number]]s, the Brahmagupta–Fibonacci identity is equivalent to the [[multiplicative function|multiplicative property]] for absolute values of [[complex numbers]]:

:<math>  | a+bi | \cdot | c+di | = | (a+bi)(c+di) | .</math>

This can be seen as follows: expanding the right side and squaring both sides, the multiplication property is equivalent to

:<math>  | a+bi |^2 \cdot | c+di |^2 = | (ac-bd)+i(ad+bc) |^2,</math>

and by the definition of absolute value this is in turn equivalent to

:<math>  (a^2+b^2)\cdot (c^2+d^2)= (ac-bd)^2+(ad+bc)^2. </math>

An equivalent calculation in the case that the variables ''a'', ''b'', ''c'', and ''d'' are [[rational number]]s shows the identity may be interpreted as the statement that the [[field norm|norm]] in the [[field (mathematics)|field]] '''Q'''(''i'') is multiplicative: the norm is given by 
: <math>N(a+bi) = a^2 + b^2,</math>
and the multiplicativity calculation is the same as the preceding one.

== Application to Pell's equation ==
In its original context, Brahmagupta applied his discovery of this identity to the solution of [[Pell's equation]] ''x''<sup>2</sup>&nbsp;&minus;&nbsp;''Ay''<sup>2</sup>&nbsp;=&nbsp;1. Using the identity in the more general form

:<math>(x_1^2 - Ay_1^2)(x_2^2 - Ay_2^2) = (x_1x_2 + Ay_1y_2)^2 - A(x_1y_2 + x_2y_1)^2, </math>

he was able to "compose" triples (''x''<sub>1</sub>,&nbsp;''y''<sub>1</sub>,&nbsp;''k''<sub>1</sub>) and (''x''<sub>2</sub>,&nbsp;''y''<sub>2</sub>,&nbsp;''k''<sub>2</sub>) that were solutions of ''x''<sup>2</sup>&nbsp;&minus;&nbsp;''Ay''<sup>2</sup>&nbsp;=&nbsp;''k'', to generate the new triple

:<math>(x_1x_2 + Ay_1y_2 \,,\, x_1y_2 + x_2y_1 \,,\, k_1k_2).</math>

Not only did this give a way to generate infinitely many solutions to ''x''<sup>2</sup>&nbsp;&minus;&nbsp;''Ay''<sup>2</sup>&nbsp;=&nbsp;1 starting with one solution, but also, by dividing such a composition by ''k''<sub>1</sub>''k''<sub>2</sub>, integer or "nearly integer" solutions could often be obtained. The general method for solving the Pell equation given by [[Bhaskara II]] in 1150, namely the [[chakravala method|chakravala (cyclic) method]], was also based on this identity.<ref name=stillwell>{{Harvnb|Stillwell|2002|pp=72–76}}</ref>

== Writing integers as a sum of two squares ==
When used in conjunction with one of [[Fermat's theorem on sums of two squares|Fermat's theorems]], the Brahmagupta–Fibonacci identity proves that the product of a square and any number of primes of the form 4''n''&nbsp;+&nbsp;1 is a sum of two squares.

==See also==
{{Div col}}
* [[Brahmagupta matrix]]
* [[Indian mathematics]]
* [[Brahmagupta polynomials]]
* [[List of Indian mathematicians]]
*[[List of Italian mathematicians]]
* [[Sum of two squares theorem]]
{{Div col end}}

==Notes==
{{reflist|2}}

==References==
*{{citation | last=Joseph | first=George G. | author-link=George G. Joseph | year=2000 | title=The Crest of the Peacock: The Non-European Roots of Mathematics | edition=2nd | publisher=[[Princeton University Press]] | page=306 | isbn=978-0-691-00659-8 | url={{Google books|c-xT0KNJp0cC|plainurl=yes}}}}<!-- Google books links to the 3rd edition. -->
*{{citation | last=Stillwell | first=John | author-link = John Stillwell | year=2002 | title = Mathematics and its history | edition=2nd | publisher=[[Springer Science+Business Media|Springer]] | isbn=978-0-387-95336-6 | pages=72–76 | url={{Google books|WNjRrqTm62QC|page=72|plainurl=yes}}}}

==External links==
*[http://planetmath.org/SumsOfTwoSquares Brahmagupta's identity] at [[PlanetMath]]
*[http://mathworld.wolfram.com/BrahmaguptaIdentity.html Brahmagupta Identity] on [[MathWorld]]
*[http://sites.google.com/site/tpiezas/005b/  A Collection of Algebraic Identities] {{Webarchive|url=https://web.archive.org/web/20120306122543/http://sites.google.com/site/tpiezas/005b |date=2012-03-06 }}

{{DEFAULTSORT:Brahmagupta-Fibonacci identity}}
[[Category:Brahmagupta]]
[[Category:Algebraic identities]]
[[Category:Squares in number theory]]