Degen's eight-square identity is the incredible polynomial identity
 |
(1)
|
found around 1818 by the Danish mathematician Ferdinand Degen (1766-1825). It was subsequently independently rediscovered twice: in 1843 by the jurist and mathematician John Thomas Graves (1806-1870) and in 1845 by Arthur Cayley (1821-1895). Since the identity follows from the fact that the norm of the product of two octonions is the product of the norms, the octonions are sometimes known as the Cayley numbers.
Given a sum of 
squares identity of the form
 |
(2)
|
where the 
are bilinear over the independent variables 
and 
, Adolf Hurwitz proved in 1898 that such identities are possible
iff 
, 2, 4, 8. The case 
corresponds to the identity
 |
(3)
|
and the case 
corresponds to the Euler
four-square identity and 
to Degen's eight-square identity.
If the 
are just rational functions of the 
and 
, then Albrecht Pfister proved in 1967 that identities can
be found for 
any power of 2.
