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The eight-square identity is a polynomial identity of the following form
(1) |
It can be written in many forms, for instance
(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 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
(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
(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 .
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. |
[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.