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In [[algebra]], the '''Brahmagupta–Fibonacci identity''' or simply '''Fibonacci's identity''' (and in fact due to [[Diophantus|Diophantus of Alexandria]]) says that the product of two sums each of two squares is itself a sum of two squares. In other words, the set of all sums of two squares is [[closure (mathematics)|closed]] under multiplication. Specifically:
:<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 a special case (''n''&nbsp;=&nbsp;2) of [[Lagrange's identity]], and is first found in [[Diophantus]]. [[Brahmagupta]] proved and used a more general identity (the Brahmagupta identity), equivalent to
:<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>
showing that the set of all numbers of the form <math>x^2 + ny^2</math> is closed under multiplication.


Both (1) and (2) 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''.


This identity holds in both the [[integer|ring of integers]] and the [[rational number|ring of rational numbers]], and more generally in any [[commutative ring]].
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In the [[integer]] case this identity finds applications in [[number theory]] for example when used in conjunction with one of [[Fermat's theorem on sums of two squares|Fermat's theorems]] it proves that the product of a square and any number of primes of the form 4''n''&nbsp;+&nbsp;1 is also a sum of two squares.
 
==History==
The identity is actually first found in [[Diophantus]]' ''[[Arithmetica]]'' (III, 19), of the third century BC.
It was rediscovered by Brahmagupta (598&ndash;668), an [[Indian mathematicians|Indian mathematician]] and [[Indian astronomy|astronomer]], who generalized it (to the Brahmagupta 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>George G. Joseph (2000). ''The Crest of the Peacock'', p. 306. [[Princeton University Press]]. ISBN 0-691-00659-8.</ref> The identity later appeared in [[Fibonacci]]'s ''[[The Book of Squares|Book of Squares]]'' in 1225.
 
==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|Cayley numbers]] which has connections to [[Bott periodicity]]. There is also [[Pfister's sixteen-square identity]], though it is no longer bilinear.
 
== Relation to complex numbers ==
 
If ''a'', ''b'', ''c'', and ''d'' are [[real number]]s, this identity is equivalent to the multiplication property for absolute values of [[complex numbers]] namely that:
 
:<math>  | a+bi |  | c+di | = | (a+bi)(c+di) | \,</math>
 
since
 
:<math> | a+bi |  | c+di | = | (ac-bd)+i(ad+bc) |,\,</math>
 
by squaring both sides
 
:<math>  | a+bi |^2 | c+di |^2 = | (ac-bd)+i(ad+bc) |^2,\,</math>
 
and by the definition of absolute value,
 
:<math>  (a^2+b^2)(c^2+d^2)= (ac-bd)^2+(ad+bc)^2. \,</math>
 
== Interpretation via norms ==
 
In the case that the variables ''a'', ''b'', ''c'', and ''d'' are [[rational number]]s, the identity may be interpreted as the statement that the [[field norm|norm]] in the [[field (mathematics)|field]] '''Q'''(''i'') is ''multiplicative''. That is, we have
 
: <math>N(a+bi) = a^2 + b^2 \text{ and }N(c+di) = c^2 + d^2, \,</math>
 
and also
 
: <math>N((a+bi)(c+di)) = N((ac-bd)+i(ad+bc)) = (ac-bd)^2 + (ad+bc)^2. \,</math>
 
Therefore the identity is saying that
 
: <math>N((a+bi)(c+di)) = N(a+bi) \cdot N(c+di). \,</math>
 
== Application to Pell's equation ==
In its original context, Brahmagupta applied his discovery (the [[Brahmagupta identity]]) to the solution of [[Pell's equation]], namely ''x''<sup>2</sup>&nbsp;&minus;&nbsp;''Ny''<sup>2</sup>&nbsp;=&nbsp;1. Using the identity in the more general form
 
:<math>(x_1^2 - Ny_1^2)(x_2^2 - Ny_2^2) = (x_1x_2 + Ny_1y_2)^2 - N(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;''Ny''<sup>2</sup>&nbsp;=&nbsp;''k'', to generate the new triple
 
:<math>(x_1x_2 + Ny_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;''Ny''<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>{{citation | year=2002 | title = Mathematics and its history | author1=[[John Stillwell]] | edition=2 | publisher=Springer | isbn=978-0-387-95336-6 | pages=72–76 | url=http://books.google.com/books?id=WNjRrqTm62QC&pg=PA72}}</ref>
 
==See also==
* [[Brahmagupta matrix]]
* [[Indian mathematics]]
* [[List of Indian mathematicians]]
* [[Euler's four-square identity]]
 
==References==
{{reflist}}
 
==External links==
*[http://planetmath.org/encyclopedia/BrahmaguptasIdentity.html 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]
 
{{DEFAULTSORT:Brahmagupta-Fibonacci identity}}
[[Category:Algebra]]
[[Category:Elementary algebra]]
[[Category:Mathematical identities]]
[[Category:Brahmagupta]]

Latest revision as of 21:49, 10 December 2014


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