Pfister's Sixteen-square Identity

In algebra, Pfister's sixteen-square identity is a non- bilinear identity of form

$\left(x_1^2+x_2^2+x_3^2+\cdots+x_^2\right)\left(y_1^2+y_2^2+y_3^2+\cdots+y_^2\right) = z_1^2+z_2^2+z_3^2+\cdots+z_^2$

It was first proven to exist by H. Zassenhaus and W. Eichhorn in the 1960s, and independently by Albrecht Pfister around the same time. There are several versions, a concise one of which is

$\begin &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \end$

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 $u_i$ are

$\begin &u_1 = \tfrac \\ &u_2 = \tfrac \\ &u_3 = \tfrac \\ &u_4 = \tfrac \\ &u_5 = \tfrac \\ &u_6 = \tfrac \\ &u_7 = \tfrac \\ &u_8 = \tfrac \end$

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,

$u_1^2+u_2^2+u_3^2+u_4^2+u_5^2+u_6^2+u_7^2+u_8^2 = x_^2+x_^2+x_^2+x_^2+x_^2+x_^2+x_^2+x_^2$

No sixteen-square identity exists involving only bilinear functions since Hurwitz's theorem states an identity of the form

$\left(x_1^2+x_2^2+x_3^2+\cdots+x_n^2)(y_1^2+y_2^2+y_3^2+\cdots+y_n^2\right) = z_1^2+z_2^2+z_3^2+\cdots+z_n^2$

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 $n = 2^m$.Pfister's Theorem on Sums of Squares, Keith Conrad, http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/pfister.pdf 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

External links

Pfister's 16-Square Identity

Analytic number theory
Algebraic identities
Sedenions