In algebra, Pfister's sixteen-square identity is a non-bilinear identity of form
\( {\displaystyle (x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+\cdots +x_{16}^{2})\,(y_{1}^{2}+y_{2}^{2}+y_{3}^{2}+\cdots +y_{16}^{2})=z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+\cdots +z_{16}^{2}} \)
It was first proven to exist by H. Zassenhaus and W. Eichhorn in the 1960s,[1] and independently by Pfister[2] around the same time. There are several versions, a concise one of which is
\( \,^{{z_{1}={\color {blue}{x_{1}y_{1}-x_{2}y_{2}-x_{3}y_{3}-x_{4}y_{4}-x_{5}y_{5}-x_{6}y_{6}-x_{7}y_{7}-x_{8}y_{8}}}+u_{1}y_{9}-u_{2}y_{{10}}-u_{3}y_{{11}}-u_{4}y_{{12}}-u_{5}y_{{13}}-u_{6}y_{{14}}-u_{7}y_{{15}}-u_{8}y_{{16}}}} \)
\( \,^{{z_{2}={\color {blue}{x_{2}y_{1}+x_{1}y_{2}+x_{4}y_{3}-x_{3}y_{4}+x_{6}y_{5}-x_{5}y_{6}-x_{8}y_{7}+x_{7}y_{8}}}+u_{2}y_{9}+u_{1}y_{{10}}+u_{4}y_{{11}}-u_{3}y_{{12}}+u_{6}y_{{13}}-u_{5}y_{{14}}-u_{8}y_{{15}}+u_{7}y_{{16}}}} \)
\( \,^{{z_{3}={\color {blue}{x_{3}y_{1}-x_{4}y_{2}+x_{1}y_{3}+x_{2}y_{4}+x_{7}y_{5}+x_{8}y_{6}-x_{5}y_{7}-x_{6}y_{8}}}+u_{3}y_{9}-u_{4}y_{{10}}+u_{1}y_{{11}}+u_{2}y_{{12}}+u_{7}y_{{13}}+u_{8}y_{{14}}-u_{5}y_{{15}}-u_{6}y_{{16}}}} \)
\( \,^{{z_{4}={\color {blue}{x_{4}y_{1}+x_{3}y_{2}-x_{2}y_{3}+x_{1}y_{4}+x_{8}y_{5}-x_{7}y_{6}+x_{6}y_{7}-x_{5}y_{8}}}+u_{4}y_{9}+u_{3}y_{{10}}-u_{2}y_{{11}}+u_{1}y_{{12}}+u_{8}y_{{13}}-u_{7}y_{{14}}+u_{6}y_{{15}}-u_{5}y_{{16}}}} \)
\( \,^{{z_{5}={\color {blue}{x_{5}y_{1}-x_{6}y_{2}-x_{7}y_{3}-x_{8}y_{4}+x_{1}y_{5}+x_{2}y_{6}+x_{3}y_{7}+x_{4}y_{8}}}+u_{5}y_{9}-u_{6}y_{{10}}-u_{7}y_{{11}}-u_{8}y_{{12}}+u_{1}y_{{13}}+u_{2}y_{{14}}+u_{3}y_{{15}}+u_{4}y_{{16}}}} \)
\( \,^{{z_{6}={\color {blue}{x_{6}y_{1}+x_{5}y_{2}-x_{8}y_{3}+x_{7}y_{4}-x_{2}y_{5}+x_{1}y_{6}-x_{4}y_{7}+x_{3}y_{8}}}+u_{6}y_{9}+u_{5}y_{{10}}-u_{8}y_{{11}}+u_{7}y_{{12}}-u_{2}y_{{13}}+u_{1}y_{{14}}-u_{4}y_{{15}}+u_{3}y_{{16}}}} \)
\( \,^{{z_{7}={\color {blue}{x_{7}y_{1}+x_{8}y_{2}+x_{5}y_{3}-x_{6}y_{4}-x_{3}y_{5}+x_{4}y_{6}+x_{1}y_{7}-x_{2}y_{8}}}+u_{7}y_{9}+u_{8}y_{{10}}+u_{5}y_{{11}}-u_{6}y_{{12}}-u_{3}y_{{13}}+u_{4}y_{{14}}+u_{1}y_{{15}}-u_{2}y_{{16}}}} \)
\( \,^{{z_{8}={\color {blue}{x_{8}y_{1}-x_{7}y_{2}+x_{6}y_{3}+x_{5}y_{4}-x_{4}y_{5}-x_{3}y_{6}+x_{2}y_{7}+x_{1}y_{8}}}+u_{8}y_{9}-u_{7}y_{{10}}+u_{6}y_{{11}}+u_{5}y_{{12}}-u_{4}y_{{13}}-u_{3}y_{{14}}+u_{2}y_{{15}}+u_{1}y_{{16}}}} \)
\( \,^{{z_{9}=x_{9}y_{1}-x_{{10}}y_{2}-x_{{11}}y_{3}-x_{{12}}y_{4}-x_{{13}}y_{5}-x_{{14}}y_{6}-x_{{15}}y_{7}-x_{{16}}y_{8}+x_{1}y_{9}-x_{2}y_{{10}}-x_{3}y_{{11}}-x_{4}y_{{12}}-x_{5}y_{{13}}-x_{6}y_{{14}}-x_{7}y_{{15}}-x_{8}y_{{16}}}} \)
\( \,^{{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}}}}
\( \,^{{z_{{11}}=x_{{11}}y_{1}-x_{{12}}y_{2}+x_{9}y_{3}+x_{{10}}y_{4}+x_{{15}}y_{5}+x_{{16}}y_{6}-x_{{13}}y_{7}-x_{{14}}y_{8}+x_{3}y_{9}-x_{4}y_{{10}}+x_{1}y_{{11}}+x_{2}y_{{12}}+x_{7}y_{{13}}+x_{8}y_{{14}}-x_{5}y_{{15}}-x_{6}y_{{16}}}} \)
\( \,^{{z_{{12}}=x_{{12}}y_{1}+x_{{11}}y_{2}-x_{{10}}y_{3}+x_{9}y_{4}+x_{{16}}y_{5}-x_{{15}}y_{6}+x_{{14}}y_{7}-x_{{13}}y_{8}+x_{4}y_{9}+x_{3}y_{{10}}-x_{2}y_{{11}}+x_{1}y_{{12}}+x_{8}y_{{13}}-x_{7}y_{{14}}+x_{6}y_{{15}}-x_{5}y_{{16}}}} \)
\( \,^{{z_{{13}}=x_{{13}}y_{1}-x_{{14}}y_{2}-x_{{15}}y_{3}-x_{{16}}y_{4}+x_{9}y_{5}+x_{{10}}y_{6}+x_{{11}}y_{7}+x_{{12}}y_{8}+x_{5}y_{9}-x_{6}y_{{10}}-x_{7}y_{{11}}-x_{8}y_{{12}}+x_{1}y_{{13}}+x_{2}y_{{14}}+x_{3}y_{{15}}+x_{4}y_{{16}}}} \)
\( \,^{{z_{{14}}=x_{{14}}y_{1}+x_{{13}}y_{2}-x_{{16}}y_{3}+x_{{15}}y_{4}-x_{{10}}y_{5}+x_{9}y_{6}-x_{{12}}y_{7}+x_{{11}}y_{8}+x_{6}y_{9}+x_{5}y_{{10}}-x_{8}y_{{11}}+x_{7}y_{{12}}-x_{2}y_{{13}}+x_{1}y_{{14}}-x_{4}y_{{15}}+x_{3}y_{{16}}}} \)
\( \,^{{z_{{15}}=x_{{15}}y_{1}+x_{{16}}y_{2}+x_{{13}}y_{3}-x_{{14}}y_{4}-x_{{11}}y_{5}+x_{{12}}y_{6}+x_{9}y_{7}-x_{{10}}y_{8}+x_{7}y_{9}+x_{8}y_{{10}}+x_{5}y_{{11}}-x_{6}y_{{12}}-x_{3}y_{{13}}+x_{4}y_{{14}}+x_{1}y_{{15}}-x_{2}y_{{16}}}} \)
\( \,^{{z_{{16}}=x_{{16}}y_{1}-x_{{15}}y_{2}+x_{{14}}y_{3}+x_{{13}}y_{4}-x_{{12}}y_{5}-x_{{11}}y_{6}+x_{{10}}y_{7}+x_{9}y_{8}+x_{8}y_{9}-x_{7}y_{{10}}+x_{6}y_{{11}}+x_{5}y_{{12}}-x_{4}y_{{13}}-x_{3}y_{{14}}+x_{2}y_{{15}}+x_{1}y_{{16}}}} \)
If all \( x_{i} \) and \( y_{i} \) with i>8 are set equal to zero, then it reduces to Degen's eight-square identity (in blue). The u i {\displaystyle u_{i}} u_{i} are
\( u_{1}={\tfrac {(ax_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2})x_{9}-2x_{1}(bx_{1}x_{9}+x_{2}x_{{10}}+x_{3}x_{{11}}+x_{4}x_{{12}}+x_{5}x_{{13}}+x_{6}x_{{14}}+x_{7}x_{{15}}+x_{8}x_{{16}})}{c}} \)
\( u_{2}={\tfrac {(x_{1}^{2}+ax_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2})x_{{10}}-2x_{2}(x_{1}x_{9}+bx_{2}x_{{10}}+x_{3}x_{{11}}+x_{4}x_{{12}}+x_{5}x_{{13}}+x_{6}x_{{14}}+x_{7}x_{{15}}+x_{8}x_{{16}})}{c}} \)
\( u_{3}={\tfrac {(x_{1}^{2}+x_{2}^{2}+ax_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2})x_{{11}}-2x_{3}(x_{1}x_{9}+x_{2}x_{{10}}+bx_{3}x_{{11}}+x_{4}x_{{12}}+x_{5}x_{{13}}+x_{6}x_{{14}}+x_{7}x_{{15}}+x_{8}x_{{16}})}{c}} \)
\( u_{4}={\tfrac {(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+ax_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2})x_{{12}}-2x_{4}(x_{1}x_{9}+x_{2}x_{{10}}+x_{3}x_{{11}}+bx_{4}x_{{12}}+x_{5}x_{{13}}+x_{6}x_{{14}}+x_{7}x_{{15}}+x_{8}x_{{16}})}{c}} \)
\( u_{5}={\tfrac {(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+ax_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2})x_{{13}}-2x_{5}(x_{1}x_{9}+x_{2}x_{{10}}+x_{3}x_{{11}}+x_{4}x_{{12}}+bx_{5}x_{{13}}+x_{6}x_{{14}}+x_{7}x_{{15}}+x_{8}x_{{16}})}{c}} \)
\( u_{6}={\tfrac {(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+ax_{6}^{2}+x_{7}^{2}+x_{8}^{2})x_{{14}}-2x_{6}(x_{1}x_{9}+x_{2}x_{{10}}+x_{3}x_{{11}}+x_{4}x_{{12}}+x_{5}x_{{13}}+bx_{6}x_{{14}}+x_{7}x_{{15}}+x_{8}x_{{16}})}{c}} \)
\( u_{7}={\tfrac {(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+ax_{7}^{2}+x_{8}^{2})x_{{15}}-2x_{7}(x_{1}x_{9}+x_{2}x_{{10}}+x_{3}x_{{11}}+x_{4}x_{{12}}+x_{5}x_{{13}}+x_{6}x_{{14}}+bx_{7}x_{{15}}+x_{8}x_{{16}})}{c}} \)
\( u_{8}={\tfrac {(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+ax_{8}^{2})x_{{16}}-2x_{8}(x_{1}x_{9}+x_{2}x_{{10}}+x_{3}x_{{11}}+x_{4}x_{{12}}+x_{5}x_{{13}}+x_{6}x_{{14}}+x_{7}x_{{15}}+bx_{8}x_{{16}})}{c}} \)
and,
\( a=-1,\;\;b=0,\;\;c=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}\,. \)
The identity shows that, in general, the product of two sums of sixteen squares is the sum of sixteen rational squares. Incidentally, the \( u_{i} \) also obey,
\( {\displaystyle u_{1}^{2}+u_{2}^{2}+u_{3}^{2}+u_{4}^{2}+u_{5}^{2}+u_{6}^{2}+u_{7}^{2}+u_{8}^{2}=x_{9}^{2}+x_{10}^{2}+x_{11}^{2}+x_{12}^{2}+x_{13}^{2}+x_{14}^{2}+x_{15}^{2}+x_{16}^{2}} \)
No sixteen-square identity exists involving only bilinear functions since Hurwitz's theorem states an identity of the form
\({\displaystyle (x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+\cdots +x_{n}^{2})(y_{1}^{2}+y_{2}^{2}+y_{3}^{2}+\cdots +y_{n}^{2})=z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+\cdots +z_{n}^{2}} \)
with the \( z_{i} \) bilinear functions of the \( x_{i} \) and \( y_{i} \) is possible only for n ∈ {1, 2, 4, 8} . However, the more general Pfister's theorem (1965) shows that if the \( z_{i} \) are rational functions of one set of variables, hence has a denominator, then it is possible for all \( n=2^{m} \).[3] There are also non-bilinear versions of Euler's four-square and Degen's eight-square identities.
See also
Brahmagupta–Fibonacci identity
Euler's four-square identity
Degen's eight-square identity
Sedenions
References
H. Zassenhaus and W. Eichhorn, "Herleitung von Acht- und Sechzehn-Quadrate-Identitäten mit Hilfe von Eigenschaften der verallgemeinerten Quaternionen und der Cayley-Dicksonchen Zahlen," Arch. Math. 17 (1966), 492-496
A. Pfister, Zur Darstellung von -1 als Summe von Quadraten in einem Körper," J. London Math. Soc. 40 (1965), 159-165
Pfister's Theorem on Sums of Squares, Keith Conrad, http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/pfister.pdf
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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