Pfister's sixteen-square identity explained

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

$$\backslash left(x\_1^2+x\_2^2+x\_3^2+\backslash cdots+x\_^2\backslash right)\backslash left(y\_1^2+y\_2^2+y\_3^2+\backslash cdots+y\_^2\backslash right)\; =\; z\_1^2+z\_2^2+z\_3^2+\backslash cdots+z\_^2$$

It was first proven to exist by H. Zassenhaus and W. Eichhorn in the 1960s,^[1] and independently by Albrecht Pfister^[2] around the same time. There are several versions, a concise one of which is

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

and,

$$a=-1,\backslash ;\backslash ;b=0,\backslash ;\backslash ;c=x\_1^2+x\_2^2+x\_3^2+x\_4^2+x\_5^2+x\_6^2+x\_7^2+x\_8^2\backslash ,.$$

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

$$\backslash left(x\_1^2+x\_2^2+x\_3^2+\backslash cdots+x\_n^2)(y\_1^2+y\_2^2+y\_3^2+\backslash cdots+y\_n^2\backslash right)\; =\; z\_1^2+z\_2^2+z\_3^2+\backslash cdots+z\_n^2$$

with the

z _{i} bilinear functions of the x _{i} and y _{i} is possible only for n ∈ . 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

1. 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
2. A. Pfister, Zur Darstellung von -1 als Summe von Quadraten in einem Körper," J. London Math. Soc. 40 (1965), 159-165
3. Pfister's Theorem on Sums of Squares, Keith Conrad, http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/pfister.pdf

External links

• Pfister's 16-Square Identity