|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| In [[mathematics]], '''Euler's four-square identity''' says that the product of two numbers, each of which is a sum of four [[Square (algebra)|square]]s, is itself a sum of four squares. Specifically:
| |
|
| |
|
| :<math>(a_1^2+a_2^2+a_3^2+a_4^2)(b_1^2+b_2^2+b_3^2+b_4^2)=\,</math>
| |
|
| |
|
| ::<math>(a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4)^2 +\,</math>
| | Material to meet you! Some of my name is Eusebio also I think it music quite good when we say it. My your own house is now in Vermont and I don't program on changing it. [http://www.squidoo.com/search/results?q=Software+building Software building] up has been my day job for a but. To farrenheit is the only hobby my wife doesn't agree to. I'm not good at [http://browse.deviantart.com/?qh=§ion=&global=1&q=webdesign webdesign] but you might love to check my website: http://prometeu.net<br><br>Also visit my homepage: [http://prometeu.net clash of Clans Hack download no survey] |
| | |
| ::<math>(a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3)^2 +\,</math>
| |
| | |
| ::<math>(a_1 b_3 - a_2 b_4 + a_3 b_1 + a_4 b_2)^2 +\,</math>
| |
| | |
| ::<math>(a_1 b_4 + a_2 b_3 - a_3 b_2 + a_4 b_1)^2.\,</math>
| |
| | |
| [[Leonhard Euler|Euler]] wrote about this identity in a letter dated May 4, 1748 to [[Christian Goldbach|Goldbach]]<ref>''Leonhard Euler: Life, Work and Legacy'', R.E. Bradley and C.E. Sandifer (eds), Elsevier, 2007, p. 193</ref><ref>''Mathematical Evolutions'', A. Shenitzer and J. Stillwell (eds), Math. Assoc. America, 2002, p. 174</ref> (but he used a different sign convention from the above). It can be proven with [[elementary algebra]] and holds in every [[commutative ring]]. If the <math>a_k</math> and <math>b_k</math> are [[real number]]s, a more elegant proof is available: the identity expresses the fact that the absolute value of the product of two [[quaternion]]s is equal to the product of their absolute values, in the same way that the [[Brahmagupta–Fibonacci identity|Brahmagupta–Fibonacci two-square identity]] does for [[complex numbers]].
| |
| | |
| The identity was used by [[Joseph Louis Lagrange|Lagrange]] to prove his [[Lagrange's four-square theorem|four square theorem]]. More specifically, it implies that it is sufficient to prove the theorem for [[prime numbers]], after which the more general theorem follows. The sign convention used above corresponds to the signs obtained by multiplying two quaternions. Other sign conventions can be obtained by changing any <math>a_k</math> to <math>-a_k</math>, <math>b_k</math> to <math>-b_k</math>, or by changing the signs inside any of the squared terms on the right hand side.
| |
| | |
| [[Hurwitz's theorem (normed division algebras)|Hurwitz's theorem]] states that an identity of form,
| |
| | |
| :<math>(a_1^2+a_2^2+a_3^2+...+a_n^2)(b_1^2+b_2^2+b_3^2+...+b_n^2) = c_1^2+c_2^2+c_3^2+...+c_n^2\,</math>
| |
| | |
| where the <math>c_i</math> are [[bilinear map|bilinear]] functions of the <math>a_i</math> and <math>b_i</math> is possible only for ''n'' = {1, 2, 4, 8}. However, the more general [[Pfister's theorem]] allows that if the <math>c_i</math> are just [[rational functions]] of one set of variables, hence has a [[denominator]], then it is possible for all <math>n = 2^m</math>.<ref>Pfister's Theorem on Sums of Squares, Keith Conrad, http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/pfister.pdf</ref> Thus, a different kind of four-square identity can be given as,
| |
| | |
| :<math>(a_1^2+a_2^2+a_3^2+a_4^2)(b_1^2+b_2^2+b_3^2+b_4^2)=\,</math>
| |
| | |
| ::<math>(a_1 b_4 + a_2 b_3 + a_3 b_2 + a_4 b_1)^2 +\,</math> | |
| | |
| ::<math>(a_1 b_3 - a_2 b_4 + a_3 b_1 - a_4 b_2)^2 +\,</math>
| |
| | |
| ::<math>\left(a_1 b_2 + a_2 b_1 + \frac{a_3 u_1}{b_1^2+b_2^2} - \frac{a_4 u_2}{b_1^2+b_2^2}\right)^2+\,</math>
| |
| | |
| ::<math>\left(a_1 b_1 - a_2 b_2 - \frac{a_4 u_1}{b_1^2+b_2^2} - \frac{a_3 u_2}{b_1^2+b_2^2}\right)^2\,</math>
| |
| | |
| where,
| |
| | |
| :<math>u_1 = b_1^2b_4-2b_1b_2b_3-b_2^2b_4</math>
| |
| | |
| :<math>u_2 = b_1^2b_3+2b_1b_2b_4-b_2^2b_3</math>
| |
| | |
| Note also the incidental fact that,
| |
| | |
| :<math>u_1^2+u_2^2 = (b_1^2+b_2^2)^2(b_3^2+b_4^2)</math> | |
| | |
| ==See also==
| |
| * [[Degen's eight-square identity]]
| |
| * [[Pfister's sixteen-square identity]]
| |
| * [[Latin square]]
| |
| | |
| ==References==
| |
| <references/>
| |
| | |
| ==External links==
| |
| *[http://sites.google.com/site/tpiezas/005b/ A Collection of Algebraic Identities]
| |
| *[http://math.dartmouth.edu/~euler/correspondence/letters/OO0841.pdf] Lettre CXV from Euler to Goldbach
| |
| | |
| {{DEFAULTSORT:Euler's Four-Square Identity}}
| |
| [[Category:Elementary algebra]]
| |
| [[Category:Elementary number theory]]
| |
| [[Category:Mathematical identities]]
| |
Material to meet you! Some of my name is Eusebio also I think it music quite good when we say it. My your own house is now in Vermont and I don't program on changing it. Software building up has been my day job for a but. To farrenheit is the only hobby my wife doesn't agree to. I'm not good at webdesign but you might love to check my website: http://prometeu.net
Also visit my homepage: clash of Clans Hack download no survey