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In [[algebra]], '''Pfister's sixteen-square identity''' is a non-[[bilinear map|bilinear]] identity of form
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:<math>
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\begin{align}
& {}\quad (x_1^2+x_2^2+x_3^2+x_4^2+\cdots+x_{16}^2)(y_1^2+y_2^2+y_3^2+y_4^2+\cdots+y_{16}^2) \\[8pt]
& = z_1^2+z_2^2+z_3^2+z_4^2+\cdots+z_{16}^2
\end{align}
</math>
 
It was first proven to exist by [[Hans Zassenhaus|H. Zassenhaus]] and W. Eichhorn in the 1960s,<ref>H. Zassenhaus and W. Eichhorn, "Herleitung von Acht- und Sechzehn-Quadrate-Identitaten mit Hilfe von Eigenschaften der verallgemeinerten Quaternionen und der Cayley-Dicksonchen Zahlen," Arch. Math. 17 (1966), 492-496</ref> and independently by Pfister<ref>A. Pfister, Zur Darstellung von -1 als Summe von Quadraten in einem Korper," J. London Math. Soc. 40 (1965), 159-165</ref> around the same time.  There are several versions, a concise one of which is
 
:<math>\,^{z_1 = x_1 y_1 - x_2 y_2 - x_3 y_3 - x_4 y_4 - x_5 y_5 - x_6 y_6 - x_7 y_7 - x_8 y_8 + u_1 y_9 - u_2 y_{10} - u_3 y_{11} - u_4 y_{12} - u_5 y_{13} - u_6 y_{14} - u_7 y_{15} - u_8 y_{16}}</math>
:<math>\,^{z_2 = x_2 y_1 + x_1 y_2 + x_4 y_3 - x_3 y_4 + x_6 y_5 - x_5 y_6 - x_8 y_7 + x_7 y_8 + u_2 y_9 + u_1 y_{10} + u_4 y_{11} - u_3 y_{12} + u_6 y_{13} - u_5 y_{14} - u_8 y_{15} + u_7 y_{16}}</math>
:<math>\,^{z_3 = x_3 y_1 - x_4 y_2 + x_1 y_3 + x_2 y_4 + x_7 y_5 + x_8 y_6 - x_5 y_7 - x_6 y_8 + u_3 y_9 - u_4 y_{10} + u_1 y_{11} + u_2 y_{12} + u_7 y_{13} + u_8 y_{14} - u_5 y_{15} - u_6 y_{16}}</math>
:<math>\,^{z_4 = x_4 y_1 + x_3 y_2 - x_2 y_3 + x_1 y_4 + x_8 y_5 - x_7 y_6 + x_6 y_7 - x_5 y_8 + u_4 y_9 + u_3 y_{10} - u_2 y_{11} + u_1 y_{12} + u_8 y_{13} - u_7 y_{14} + u_6 y_{15} - u_5 y_{16}}</math>
:<math>\,^{z_5 = x_5 y_1 - x_6 y_2 - x_7 y_3 - x_8 y_4 + x_1 y_5 + x_2 y_6 + x_3 y_7 + x_4 y_8 + u_5 y_9 - u_6 y_{10} - u_7 y_{11} - u_8 y_{12} + u_1 y_{13} + u_2 y_{14} + u_3 y_{15} + u_4 y_{16}}</math>
:<math>\,^{z_6 = x_6 y_1 + x_5 y_2 - x_8 y_3 + x_7 y_4 - x_2 y_5 + x_1 y_6 - x_4 y_7 + x_3 y_8 + u_6 y_9 + u_5 y_{10} - u_8 y_{11} + u_7 y_{12} - u_2 y_{13} + u_1 y_{14} - u_4 y_{15} + u_3 y_{16}}</math>
:<math>\,^{z_7 = x_7 y_1 + x_8 y_2 + x_5 y_3 - x_6 y_4 - x_3 y_5 + x_4 y_6 + x_1 y_7 - x_2 y_8 + u_7 y_9 + u_8 y_{10} + u_5 y_{11} - u_6 y_{12} - u_3 y_{13} + u_4 y_{14} + u_1 y_{15} - u_2 y_{16}}</math>
:<math>\,^{z_8 = x_8 y_1 - x_7 y_2 + x_6 y_3 + x_5 y_4 - x_4 y_5 - x_3 y_6 + x_2 y_7 + x_1 y_8 + u_8 y_9 - u_7 y_{10} + u_6 y_{11} + u_5 y_{12} - u_4 y_{13} - u_3 y_{14} + u_2 y_{15} + u_1 y_{16}}</math>
:<math>\,^{z_9  =  x_9 y_1 - x_{10} y_2 - x_{11} y_3 - x_{12} y_4 - x_{13} y_5 - x_{14} y_6 - x_{15} y_7 - x_{16} y_8 + x_1 y_9 - x_2 y_{10} - x_3 y_{11} - x_4 y_{12} - x_5 y_{13} - x_6 y_{14} - x_7 y_{15} - x_8 y_{16}}</math>
:<math>\,^{z_{10} = x_{10} y_1 + x_9 y_2 + x_{12} y_3 - x_{11} y_4 + x_{14} y_5 - x_{13} y_6 - x_{16} y_7 + x_{15} y_8 + x_2 y_9 + x_1 y_{10} + x_4 y_{11} - x_3 y_{12} + x_6 y_{13} - x_5 y_{14} - x_8 y_{15} + x_7 y_{16}}</math>
:<math>\,^{z_{11} = x_{11} y_1 - x_{12} y_2 + x_9 y_3 + x_{10} y_4 + x_{15} y_5 + x_{16} y_6 - x_{13} y_7 - x_{14} y_8 + x_3 y_9 - x_4 y_{10} + x_1 y_{11} + x_2 y_{12} + x_7 y_{13} + x_8 y_{14} - x_5 y_{15} - x_6 y_{16}}</math>
:<math>\,^{z_{12} = x_{12} y_1 + x_{11} y_2 - x_{10} y_3 + x_9 y_4 + x_{16} y_5 - x_{15} y_6 + x_{14} y_7 - x_{13} y_8 + x_4 y_9 + x_3 y_{10} - x_2 y_{11} + x_1 y_{12} + x_8 y_{13} - x_7 y_{14} + x_6 y_{15} - x_5 y_{16}}</math>
:<math>\,^{z_{13} = x_{13} y_1 - x_{14} y_2 - x_{15} y_3 - x_{16} y_4 + x_9 y_5 + x_{10} y_6 + x_{11} y_7 + x_{12} y_8 + x_5 y_9 - x_6 y_{10} - x_7 y_{11} - x_8 y_{12} + x_1 y_{13} + x_2 y_{14} + x_3 y_{15} + x_4 y_{16}}</math>
:<math>\,^{z_{14} = x_{14} y_1 + x_{13} y_2 - x_{16} y_3 + x_{15} y_4 - x_{10} y_5 + x_9 y_6 - x_{12} y_7 + x_{11} y_8 + x_6 y_9 + x_5 y_{10} - x_8 y_{11} + x_7 y_{12} - x_2 y_{13} + x_1 y_{14} - x_4 y_{15} + x_3 y_{16}}</math>
:<math>\,^{z_{15} = x_{15} y_1 + x_{16} y_2 + x_{13} y_3 - x_{14} y_4 - x_{11} y_5 + x_{12} y_6 + x_9 y_7 - x_{10} y_8 + x_7 y_9 + x_8 y_{10} + x_5 y_{11} - x_6 y_{12} - x_3 y_{13} + x_4 y_{14} + x_1 y_{15} - x_2 y_{16}}</math>
:<math>\,^{z_{16} = x_{16} y_1 - x_{15} y_2 + x_{14} y_3 + x_{13} y_4 - x_{12} y_5 - x_{11} y_6 + x_{10} y_7 + x_9 y_8 + x_8 y_9 - x_7 y_{10} + x_6 y_{11} + x_5 y_{12} - x_4 y_{13} - x_3 y_{14} + x_2 y_{15} + x_1 y_{16}}</math>
 
where the <math>u_i</math> are,
 
:<math>u_1 = \tfrac{(ax_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2+x_8^2)x_9 - 2x_1(bx_1 x_9 +x_2 x_{10} +x_3 x_{11} +x_4 x_{12} +x_5 x_{13} +x_6 x_{14} +x_7 x_{15} +x_8 x_{16})}{c}</math>
:<math>u_2 = \tfrac{(x_1^2+ax_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2+x_8^2)x_{10} - 2x_2(x_1 x_9 +bx_2 x_{10} +x_3 x_{11} +x_4 x_{12} +x_5 x_{13} +x_6 x_{14} +x_7 x_{15} +x_8 x_{16})}{c}</math>
:<math>u_3 = \tfrac{(x_1^2+x_2^2+ax_3^2+x_4^2+x_5^2+x_6^2+x_7^2+x_8^2)x_{11} - 2x_3(x_1 x_9 +x_2 x_{10} +bx_3 x_{11} +x_4 x_{12} +x_5 x_{13} +x_6 x_{14} +x_7 x_{15} +x_8 x_{16})}{c}</math>
:<math>u_4 = \tfrac{(x_1^2+x_2^2+x_3^2+ax_4^2+x_5^2+x_6^2+x_7^2+x_8^2)x_{12} - 2x_4(x_1 x_9 +x_2 x_{10} +x_3 x_{11} +bx_4 x_{12} +x_5 x_{13} +x_6 x_{14} +x_7 x_{15} +x_8 x_{16})}{c}</math>
:<math>u_5 = \tfrac{(x_1^2+x_2^2+x_3^2+x_4^2+ax_5^2+x_6^2+x_7^2+x_8^2)x_{13} - 2x_5(x_1 x_9 +x_2 x_{10} +x_3 x_{11} +x_4 x_{12} +bx_5 x_{13} +x_6 x_{14} +x_7 x_{15} +x_8 x_{16})}{c}</math>
:<math>u_6 = \tfrac{(x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+ax_6^2+x_7^2+x_8^2)x_{14} - 2x_6(x_1 x_9 +x_2 x_{10} +x_3 x_{11} +x_4 x_{12} +x_5 x_{13} +bx_6 x_{14} +x_7 x_{15} +x_8 x_{16})}{c}</math>
:<math>u_7 = \tfrac{(x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+ax_7^2+x_8^2)x_{15} - 2x_7(x_1 x_9 +x_2 x_{10} +x_3 x_{11} +x_4 x_{12} +x_5 x_{13} +x_6 x_{14} +bx_7 x_{15} +x_8 x_{16})}{c}</math>
:<math>u_8 = \tfrac{(x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2+ax_8^2)x_{16} - 2x_8(x_1 x_9 +x_2 x_{10} +x_3 x_{11} +x_4 x_{12} +x_5 x_{13} +x_6 x_{14} +x_7 x_{15} +bx_8 x_{16})}{c}</math>
 
and,
 
:<math>a=-1,\;\;b=0,\;\;c=x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2+x_8^2\,.</math>
 
The <math>u_i</math> also obey,
 
:<math>u_1^2+u_2^2+u_3^2+u_4^2+u_5^2+u_6^2+u_7^2+u_8^2 = x_{9}^2+x_{10}^2+x_{11}^2+x_{12}^2+x_{13}^2+x_{14}^2+x_{15}^2+x_{16}^2\,</math>
 
Thus the identity shows that, in general, the product of two sums of sixteen squares is the sum of sixteen [[rational number|rational]] squares. If all <math>x_i,y_i</math> with <math>i>8</math> are set equal to zero, then it reduces to the [[Degen's eight-square identity|Degen's eight-square]].
 
No sixteen-square identity exists involving only bilinear functions since [[Hurwitz's theorem (normed division algebras)|Hurwitz's theorem]] states an identity of the form
 
:<math>(x_1^2+x_2^2+x_3^2+\cdots+x_n^2)(y_1^2+y_2^2+y_3^2+\cdots+y_n^2) = z_1^2+z_2^2+z_3^2+\cdots+z_n^2\,</math>
 
with the <math>z_i</math> [[bilinear map|bilinear]] functions of the <math>x_i</math> and <math>y_i</math> is possible only for ''n'' &isin; {1, 2, 4, 8} . However, the more general [[Pfister's theorem]] (1965)  shows that if the <math>z_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> There are also non-bilinear versions of [[Euler's four-square identity|Euler's four-square]] and [[Degen's eight-square identity|Degen's eight-square]] identities.
 
==See also==
* [[Brahmagupta–Fibonacci identity]]
* [[Euler's four-square identity]]
* [[Degen's eight-square identity]]
* [[Sedenions]]
 
==References==
<references/>
 
==External links==
*[http://sites.google.com/site/tpiezas/0021c Pfister's 16-Square Identity]
 
[[Category:Analytic number theory]]
[[Category:Mathematical identities]]

Revision as of 13:56, 13 February 2014

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