Degen's eight-square identity
In mathematics, Degen's eight-square identity establishes that the product of two numbers, each of which is a sum of eight squares, is itself the sum of eight squares.
Namely:
:formula_1

::formula_2
::formula_3
::formula_4
::formula_5
::formula_6
::formula_7
::formula_8
::formula_9

First discovered by Carl Ferdinand Degen around 1818, the identity was independently rediscovered by John Thomas Graves (1843) and Arthur Cayley (1845). The latter two derived it while working on an extension of quaternions called octonions. In algebraic terms the identity means that the norm of product of two octonions equals the product of their norms: formula_10. Similar statements are true for quaternions (Euler's four-square identity), complex numbers (the Brahmagupta–Fibonacci two-square identity) and real numbers. In 1898 Adolf Hurwitz proved that there is no similar bilinear identity for 16 squares (sedenions) or any other number of squares except for 1,2,4, and 8. However, in the 1960s, H. Zassenhaus, W. Eichhorn, and A. Pfister (independently) showed there can be a non-bilinear identity for 16 squares. 

Note that each quadrant reduces to a version of Euler's four-square identity:
:formula_11

::formula_12
::formula_13
::formula_14
::formula_15

and similarly for the other three quadrants. By Pfister's theorem, a different sort of eight-square identity can be given where the formula_16, introduced below, are non-bilinear and merely rational functions of the formula_17. Thus,

:formula_18

where,

::formula_19
::formula_20
::formula_21
::formula_22
::formula_23
::formula_24
::formula_25
::formula_26

and,

::formula_27
::formula_28
::formula_29
::formula_30

with,

::formula_31

Incidentally, the formula_32 obey the identity,

::formula_33
Q395
mathematics
Concept
Q4993582
Carl Ferdinand Degen
PER
Q940427
John Thomas Graves
PER
Q159430
Arthur Cayley
PER
Q173853
quaternion
Concept
Q743418
octonion
Concept
Q1139430
algebraic terms
Concept
Q956437
norm
Concept
Q1149225
Euler's four-square identity
Concept
Q279090
Brahmagupta–Fibonacci two-square identity
Concept
Q58809
Adolf Hurwitz
PER
Q1086961
bilinear
Concept
Q79188
sedenions
Concept
Q14916038
16 squares
UNK
Q1149225
Euler's four-square identity
Concept
None
Pfister's theorem
UNK
Q41237
rational functions
Concept
Q1187196
Degen's eight-square identity
LOC
1818-01-01T00:00:00Z^^http://www.w3.org/2001/XMLSchema#dateTime
1818
DATE
1898-01-01T00:00:00Z^^http://www.w3.org/2001/XMLSchema#dateTime
1898
DATE
1960-01-01T00:00:00Z^^http://www.w3.org/2001/XMLSchema#dateTime
1960
DATE
+1^^http://www.w3.org/2001/XMLSchema#decimal
1
NUMBER
+2^^http://www.w3.org/2001/XMLSchema#decimal
2
NUMBER
+3^^http://www.w3.org/2001/XMLSchema#decimal
3
NUMBER
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4
NUMBER
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5
NUMBER
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6
NUMBER
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7
NUMBER
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NUMBER
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9
NUMBER
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1843
NUMBER
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1845
NUMBER
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1,2
NUMBER
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NUMBER
+8^^http://www.w3.org/2001/XMLSchema#decimal
8
NUMBER
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NUMBER
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NUMBER
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NUMBER
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NUMBER
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NUMBER
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NUMBER
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NUMBER
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NUMBER
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NUMBER
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NUMBER
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20
NUMBER
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NUMBER
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NUMBER
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23
NUMBER
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24
NUMBER
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NUMBER
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26
NUMBER
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27
NUMBER
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NUMBER
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NUMBER
+30^^http://www.w3.org/2001/XMLSchema#decimal
30
NUMBER
+31^^http://www.w3.org/2001/XMLSchema#decimal
31
NUMBER
+32^^http://www.w3.org/2001/XMLSchema#decimal
32
NUMBER
+33^^http://www.w3.org/2001/XMLSchema#decimal
33
NUMBER
named after
time of discovery or invention
part of