Main Index Algebraic structures Polynomial rings Multivariate polynomials Identities
Subject Index
comment on the page

Eight-square identity

The eight-square identity is a polynomial identity of the following form

(a _ 1^2 + a _ 2^2 + a _ 3^2 + a _ 4^2 + a _ 5^2 + a _ 6^2 + a _ 7^2 + a _ 8^2) (b _ 1^2 + b _ ... _ 7 b _ 2 + a _ 6 b _ 3 + a _ 5 b _ 4 - a _ 4 b _ 5 - a _ 3 b _ 6 + a _ 2 b _ 7 + a _ 1 b _ 8)^2 .(1)

It can be written in many forms, for instance

(a _ 1^2 + a _ 2^2 + a _ 3^2 + a _ 4^2 + a _ 5^2 + a _ 6^2 + a _ 7^2 + a _ 8^2) (b _ 1^2 + b _ ... _ 7 b _ 2 + a _ 6 b _ 3 - a _ 5 b _ 4 - a _ 4 b _ 5 - a _ 3 b _ 6 + a _ 2 b _ 7 + a _ 1 b _ 8)^2 .(2)

(a _ 1^2 + a _ 2^2 + a _ 3^2 + a _ 4^2 + a _ 5^2 + a _ 6^2 + a _ 7^2 + a _ 8^2) (b _ 1^2 + b _ ... _ 7 b _ 2 - a _ 6 b _ 3 + a _ 5 b _ 4 + a _ 4 b _ 5 + a _ 3 b _ 6 + a _ 2 b _ 7 + a _ 1 b _ 8)^2 .(3)

This type of identities were independently discovered many times. It seems that the first was the Danish mathematician C. P. Degen (1766-1825) who discovered  [1]  identities (2) and (3) around 1818.  Degen erroneously claimed that the identity can be extended to typeset structure squares.  His paper went unnoticed. 1

In 1845 the mathematician Arthur Cayley (1821-1895)  derived an identity of this form  from the fact that the norm of the product of two octonions (or the so-called Cayley numbers) is the product of their norms  [2] ,  [3] .  Actually, the multiplication of norms of two octonions reduces to the identity

(a _ 1^2 + a _ 2^2 + a _ 3^2 + a _ 4^2 + a _ 5^2 + a _ 6^2 + a _ 7^2 + a _ 8^2) (b _ 1^2 + b _ ... a _ 7 b _ 2 - a _ 6 b _ 3 - a _ 5 b _ 4 + a _ 4 b _ 5 + a _ 3 b _ 6 - a _ 2 b _ 7 + a _ 1 b _ 8)^2(4)

Prior to Cayley in 1843 they were rediscovered also by the jurist and mathematician John Thomas Graves (1806-1870) a friend of  the Irish mathematician W.R. Hamilton. In October 1843 Hamilton wrote Graves about his discovery of quaternions . In December Graves discovered octaves, however the result was published on Graves request by Hamilton only in July 1847. Graves found two variants of eight-square identity, the identity (4) and that which differs from (4) only in the interchange of indices 7 and 8 ( [2]  , p. 164).

The eight-square identity is therefore also called the Degen-Graves-Cayley identity.

Le Besgue ( [4] , p. 65)  attributes the following identity to M.Brioschi

(a _ 1^2 + a _ 2^2 + a _ 3^2 + a _ 4^2 + a _ 5^2 + a _ 6^2 + a _ 7^2 + a _ 8^2) (b _ 1^2 + b _ ... _ 7 b _ 2 + a _ 6 b _ 3 + a _ 5 b _ 4 - a _ 4 b _ 5 - a _ 3 b _ 6 - a _ 2 b _ 7 + a _ 1 b _ 8)^2 .(5)

In 1989 Adolf Hurwitz proved that identities of this form are possible if and only if the sums of squares have  2, 4, or 8 summands .

Notes

1 Note, that it was Degen to whom 19 years old Niels Abel (1802-1829)  submitted a “solution” to the general quintic algebraic equation.  When Degen asked for a numerical example, Abel discovered a mistake in his solution, what caused to change his mind about the solvability of the general quintic in radicals, and to prove the correct opposite statement.

References

[1]  Degen, C. P. (1817-18 (1922)). Adumbratio demonstrationis theorematis arithmeticae maxime generalis. Mémoires de l'Académie Impériale des Sciences de St. Pétersbourg, 8, 207-219.

[2]  Dickson, L. E. (1919). On quaternions and their generalization and the history of the eight square theorem. Annals of Math., (2) 20, 155-171, 297.

[3]  van der Waerden, B. L. (1976). Hamilton's Discovery of Quaternions. Mathematics Magazine, 49(5), 227-234.

[4]  Le Besgue, V. A. (1862). Introduction a la théorie des nombres. Paris: Mallet-Bachelier.

Cite this web-page as:

Štefan Porubský: Eight-square identity.

Page created  .