In algebra , Pfister's sixteen-square identity is a non-bilinear identity of form
(
x
1
2
+
x
2
2
+
x
3
2
+
⋯
+
x
16
2
)
(
y
1
2
+
y
2
2
+
y
3
2
+
⋯
+
y
16
2
)
=
z
1
2
+
z
2
2
+
z
3
2
+
⋯
+
z
16
2
{\displaystyle \left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+\cdots +x_{16}^{2}\right)\left(y_{1}^{2}+y_{2}^{2}+y_{3}^{2}+\cdots +y_{16}^{2}\right)=z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+\cdots +z_{16}^{2}}
It was first proven to exist by H. Zassenhaus and W. Eichhorn in the 1960s,[1] Albrecht Pfister [2]
z
1
=
x
1
y
1
−
x
2
y
2
−
x
3
y
3
−
x
4
y
4
−
x
5
y
5
−
x
6
y
6
−
x
7
y
7
−
x
8
y
8
+
u
1
y
9
−
u
2
y
10
−
u
3
y
11
−
u
4
y
12
−
u
5
y
13
−
u
6
y
14
−
u
7
y
15
−
u
8
y
16
{\displaystyle \scriptstyle {z_{1}={\color {blue}{x_{1}y_{1}-x_{2}y_{2}-x_{3}y_{3}-x_{4}y_{4}-x_{5}y_{5}-x_{6}y_{6}-x_{7}y_{7}-x_{8}y_{8}}}+u_{1}y_{9}-u_{2}y_{10}-u_{3}y_{11}-u_{4}y_{12}-u_{5}y_{13}-u_{6}y_{14}-u_{7}y_{15}-u_{8}y_{16}}}
z
2
=
x
2
y
1
+
x
1
y
2
+
x
4
y
3
−
x
3
y
4
+
x
6
y
5
−
x
5
y
6
−
x
8
y
7
+
x
7
y
8
+
u
2
y
9
+
u
1
y
10
+
u
4
y
11
−
u
3
y
12
+
u
6
y
13
−
u
5
y
14
−
u
8
y
15
+
u
7
y
16
{\displaystyle \scriptstyle {z_{2}={\color {blue}{x_{2}y_{1}+x_{1}y_{2}+x_{4}y_{3}-x_{3}y_{4}+x_{6}y_{5}-x_{5}y_{6}-x_{8}y_{7}+x_{7}y_{8}}}+u_{2}y_{9}+u_{1}y_{10}+u_{4}y_{11}-u_{3}y_{12}+u_{6}y_{13}-u_{5}y_{14}-u_{8}y_{15}+u_{7}y_{16}}}
z
3
=
x
3
y
1
−
x
4
y
2
+
x
1
y
3
+
x
2
y
4
+
x
7
y
5
+
x
8
y
6
−
x
5
y
7
−
x
6
y
8
+
u
3
y
9
−
u
4
y
10
+
u
1
y
11
+
u
2
y
12
+
u
7
y
13
+
u
8
y
14
−
u
5
y
15
−
u
6
y
16
{\displaystyle \scriptstyle {z_{3}={\color {blue}{x_{3}y_{1}-x_{4}y_{2}+x_{1}y_{3}+x_{2}y_{4}+x_{7}y_{5}+x_{8}y_{6}-x_{5}y_{7}-x_{6}y_{8}}}+u_{3}y_{9}-u_{4}y_{10}+u_{1}y_{11}+u_{2}y_{12}+u_{7}y_{13}+u_{8}y_{14}-u_{5}y_{15}-u_{6}y_{16}}}
z
4
=
x
4
y
1
+
x
3
y
2
−
x
2
y
3
+
x
1
y
4
+
x
8
y
5
−
x
7
y
6
+
x
6
y
7
−
x
5
y
8
+
u
4
y
9
+
u
3
y
10
−
u
2
y
11
+
u
1
y
12
+
u
8
y
13
−
u
7
y
14
+
u
6
y
15
−
u
5
y
16
{\displaystyle \scriptstyle {z_{4}={\color {blue}{x_{4}y_{1}+x_{3}y_{2}-x_{2}y_{3}+x_{1}y_{4}+x_{8}y_{5}-x_{7}y_{6}+x_{6}y_{7}-x_{5}y_{8}}}+u_{4}y_{9}+u_{3}y_{10}-u_{2}y_{11}+u_{1}y_{12}+u_{8}y_{13}-u_{7}y_{14}+u_{6}y_{15}-u_{5}y_{16}}}
z
5
=
x
5
y
1
−
x
6
y
2
−
x
7
y
3
−
x
8
y
4
+
x
1
y
5
+
x
2
y
6
+
x
3
y
7
+
x
4
y
8
+
u
5
y
9
−
u
6
y
10
−
u
7
y
11
−
u
8
y
12
+
u
1
y
13
+
u
2
y
14
+
u
3
y
15
+
u
4
y
16
{\displaystyle \scriptstyle {z_{5}={\color {blue}{x_{5}y_{1}-x_{6}y_{2}-x_{7}y_{3}-x_{8}y_{4}+x_{1}y_{5}+x_{2}y_{6}+x_{3}y_{7}+x_{4}y_{8}}}+u_{5}y_{9}-u_{6}y_{10}-u_{7}y_{11}-u_{8}y_{12}+u_{1}y_{13}+u_{2}y_{14}+u_{3}y_{15}+u_{4}y_{16}}}
z
6
=
x
6
y
1
+
x
5
y
2
−
x
8
y
3
+
x
7
y
4
−
x
2
y
5
+
x
1
y
6
−
x
4
y
7
+
x
3
y
8
+
u
6
y
9
+
u
5
y
10
−
u
8
y
11
+
u
7
y
12
−
u
2
y
13
+
u
1
y
14
−
u
4
y
15
+
u
3
y
16
{\displaystyle \scriptstyle {z_{6}={\color {blue}{x_{6}y_{1}+x_{5}y_{2}-x_{8}y_{3}+x_{7}y_{4}-x_{2}y_{5}+x_{1}y_{6}-x_{4}y_{7}+x_{3}y_{8}}}+u_{6}y_{9}+u_{5}y_{10}-u_{8}y_{11}+u_{7}y_{12}-u_{2}y_{13}+u_{1}y_{14}-u_{4}y_{15}+u_{3}y_{16}}}
z
7
=
x
7
y
1
+
x
8
y
2
+
x
5
y
3
−
x
6
y
4
−
x
3
y
5
+
x
4
y
6
+
x
1
y
7
−
x
2
y
8
+
u
7
y
9
+
u
8
y
10
+
u
5
y
11
−
u
6
y
12
−
u
3
y
13
+
u
4
y
14
+
u
1
y
15
−
u
2
y
16
{\displaystyle \scriptstyle {z_{7}={\color {blue}{x_{7}y_{1}+x_{8}y_{2}+x_{5}y_{3}-x_{6}y_{4}-x_{3}y_{5}+x_{4}y_{6}+x_{1}y_{7}-x_{2}y_{8}}}+u_{7}y_{9}+u_{8}y_{10}+u_{5}y_{11}-u_{6}y_{12}-u_{3}y_{13}+u_{4}y_{14}+u_{1}y_{15}-u_{2}y_{16}}}
z
8
=
x
8
y
1
−
x
7
y
2
+
x
6
y
3
+
x
5
y
4
−
x
4
y
5
−
x
3
y
6
+
x
2
y
7
+
x
1
y
8
+
u
8
y
9
−
u
7
y
10
+
u
6
y
11
+
u
5
y
12
−
u
4
y
13
−
u
3
y
14
+
u
2
y
15
+
u
1
y
16
{\displaystyle \scriptstyle {z_{8}={\color {blue}{x_{8}y_{1}-x_{7}y_{2}+x_{6}y_{3}+x_{5}y_{4}-x_{4}y_{5}-x_{3}y_{6}+x_{2}y_{7}+x_{1}y_{8}}}+u_{8}y_{9}-u_{7}y_{10}+u_{6}y_{11}+u_{5}y_{12}-u_{4}y_{13}-u_{3}y_{14}+u_{2}y_{15}+u_{1}y_{16}}}
z
9
=
x
9
y
1
−
x
10
y
2
−
x
11
y
3
−
x
12
y
4
−
x
13
y
5
−
x
14
y
6
−
x
15
y
7
−
x
16
y
8
+
x
1
y
9
−
x
2
y
10
−
x
3
y
11
−
x
4
y
12
−
x
5
y
13
−
x
6
y
14
−
x
7
y
15
−
x
8
y
16
{\displaystyle \scriptstyle {z_{9}=x_{9}y_{1}-x_{10}y_{2}-x_{11}y_{3}-x_{12}y_{4}-x_{13}y_{5}-x_{14}y_{6}-x_{15}y_{7}-x_{16}y_{8}+x_{1}y_{9}-x_{2}y_{10}-x_{3}y_{11}-x_{4}y_{12}-x_{5}y_{13}-x_{6}y_{14}-x_{7}y_{15}-x_{8}y_{16}}}
z
10
=
x
10
y
1
+
x
9
y
2
+
x
12
y
3
−
x
11
y
4
+
x
14
y
5
−
x
13
y
6
−
x
16
y
7
+
x
15
y
8
+
x
2
y
9
+
x
1
y
10
+
x
4
y
11
−
x
3
y
12
+
x
6
y
13
−
x
5
y
14
−
x
8
y
15
+
x
7
y
16
{\displaystyle \scriptstyle {z_{10}=x_{10}y_{1}+x_{9}y_{2}+x_{12}y_{3}-x_{11}y_{4}+x_{14}y_{5}-x_{13}y_{6}-x_{16}y_{7}+x_{15}y_{8}+x_{2}y_{9}+x_{1}y_{10}+x_{4}y_{11}-x_{3}y_{12}+x_{6}y_{13}-x_{5}y_{14}-x_{8}y_{15}+x_{7}y_{16}}}
z
11
=
x
11
y
1
−
x
12
y
2
+
x
9
y
3
+
x
10
y
4
+
x
15
y
5
+
x
16
y
6
−
x
13
y
7
−
x
14
y
8
+
x
3
y
9
−
x
4
y
10
+
x
1
y
11
+
x
2
y
12
+
x
7
y
13
+
x
8
y
14
−
x
5
y
15
−
x
6
y
16
{\displaystyle \scriptstyle {z_{11}=x_{11}y_{1}-x_{12}y_{2}+x_{9}y_{3}+x_{10}y_{4}+x_{15}y_{5}+x_{16}y_{6}-x_{13}y_{7}-x_{14}y_{8}+x_{3}y_{9}-x_{4}y_{10}+x_{1}y_{11}+x_{2}y_{12}+x_{7}y_{13}+x_{8}y_{14}-x_{5}y_{15}-x_{6}y_{16}}}
z
12
=
x
12
y
1
+
x
11
y
2
−
x
10
y
3
+
x
9
y
4
+
x
16
y
5
−
x
15
y
6
+
x
14
y
7
−
x
13
y
8
+
x
4
y
9
+
x
3
y
10
−
x
2
y
11
+
x
1
y
12
+
x
8
y
13
−
x
7
y
14
+
x
6
y
15
−
x
5
y
16
{\displaystyle \scriptstyle {z_{12}=x_{12}y_{1}+x_{11}y_{2}-x_{10}y_{3}+x_{9}y_{4}+x_{16}y_{5}-x_{15}y_{6}+x_{14}y_{7}-x_{13}y_{8}+x_{4}y_{9}+x_{3}y_{10}-x_{2}y_{11}+x_{1}y_{12}+x_{8}y_{13}-x_{7}y_{14}+x_{6}y_{15}-x_{5}y_{16}}}
z
13
=
x
13
y
1
−
x
14
y
2
−
x
15
y
3
−
x
16
y
4
+
x
9
y
5
+
x
10
y
6
+
x
11
y
7
+
x
12
y
8
+
x
5
y
9
−
x
6
y
10
−
x
7
y
11
−
x
8
y
12
+
x
1
y
13
+
x
2
y
14
+
x
3
y
15
+
x
4
y
16
{\displaystyle \scriptstyle {z_{13}=x_{13}y_{1}-x_{14}y_{2}-x_{15}y_{3}-x_{16}y_{4}+x_{9}y_{5}+x_{10}y_{6}+x_{11}y_{7}+x_{12}y_{8}+x_{5}y_{9}-x_{6}y_{10}-x_{7}y_{11}-x_{8}y_{12}+x_{1}y_{13}+x_{2}y_{14}+x_{3}y_{15}+x_{4}y_{16}}}
z
14
=
x
14
y
1
+
x
13
y
2
−
x
16
y
3
+
x
15
y
4
−
x
10
y
5
+
x
9
y
6
−
x
12
y
7
+
x
11
y
8
+
x
6
y
9
+
x
5
y
10
−
x
8
y
11
+
x
7
y
12
−
x
2
y
13
+
x
1
y
14
−
x
4
y
15
+
x
3
y
16
{\displaystyle \scriptstyle {z_{14}=x_{14}y_{1}+x_{13}y_{2}-x_{16}y_{3}+x_{15}y_{4}-x_{10}y_{5}+x_{9}y_{6}-x_{12}y_{7}+x_{11}y_{8}+x_{6}y_{9}+x_{5}y_{10}-x_{8}y_{11}+x_{7}y_{12}-x_{2}y_{13}+x_{1}y_{14}-x_{4}y_{15}+x_{3}y_{16}}}
z
15
=
x
15
y
1
+
x
16
y
2
+
x
13
y
3
−
x
14
y
4
−
x
11
y
5
+
x
12
y
6
+
x
9
y
7
−
x
10
y
8
+
x
7
y
9
+
x
8
y
10
+
x
5
y
11
−
x
6
y
12
−
x
3
y
13
+
x
4
y
14
+
x
1
y
15
−
x
2
y
16
{\displaystyle \scriptstyle {z_{15}=x_{15}y_{1}+x_{16}y_{2}+x_{13}y_{3}-x_{14}y_{4}-x_{11}y_{5}+x_{12}y_{6}+x_{9}y_{7}-x_{10}y_{8}+x_{7}y_{9}+x_{8}y_{10}+x_{5}y_{11}-x_{6}y_{12}-x_{3}y_{13}+x_{4}y_{14}+x_{1}y_{15}-x_{2}y_{16}}}
z
16
=
x
16
y
1
−
x
15
y
2
+
x
14
y
3
+
x
13
y
4
−
x
12
y
5
−
x
11
y
6
+
x
10
y
7
+
x
9
y
8
+
x
8
y
9
−
x
7
y
10
+
x
6
y
11
+
x
5
y
12
−
x
4
y
13
−
x
3
y
14
+
x
2
y
15
+
x
1
y
16
{\displaystyle \scriptstyle {z_{16}=x_{16}y_{1}-x_{15}y_{2}+x_{14}y_{3}+x_{13}y_{4}-x_{12}y_{5}-x_{11}y_{6}+x_{10}y_{7}+x_{9}y_{8}+x_{8}y_{9}-x_{7}y_{10}+x_{6}y_{11}+x_{5}y_{12}-x_{4}y_{13}-x_{3}y_{14}+x_{2}y_{15}+x_{1}y_{16}}}
If all
x
i
{\displaystyle x_{i}}
y
i
{\displaystyle y_{i}}
i
>
8
{\displaystyle i>8}
Degen's eight-square identity (in blue). The
u
i
{\displaystyle u_{i}}
u
1
=
(
a
x
1
2
+
x
2
2
+
x
3
2
+
x
4
2
+
x
5
2
+
x
6
2
+
x
7
2
+
x
8
2
)
x
9
−
2
x
1
(
b
x
1
x
9
+
x
2
x
10
+
x
3
x
11
+
x
4
x
12
+
x
5
x
13
+
x
6
x
14
+
x
7
x
15
+
x
8
x
16
)
c
{\displaystyle u_{1}={\tfrac {(ax_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2})x_{9}-2x_{1}(bx_{1}x_{9}+x_{2}x_{10}+x_{3}x_{11}+x_{4}x_{12}+x_{5}x_{13}+x_{6}x_{14}+x_{7}x_{15}+x_{8}x_{16})}{c}}}
u
2
=
(
x
1
2
+
a
x
2
2
+
x
3
2
+
x
4
2
+
x
5
2
+
x
6
2
+
x
7
2
+
x
8
2
)
x
10
−
2
x
2
(
x
1
x
9
+
b
x
2
x
10
+
x
3
x
11
+
x
4
x
12
+
x
5
x
13
+
x
6
x
14
+
x
7
x
15
+
x
8
x
16
)
c
{\displaystyle u_{2}={\tfrac {(x_{1}^{2}+ax_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2})x_{10}-2x_{2}(x_{1}x_{9}+bx_{2}x_{10}+x_{3}x_{11}+x_{4}x_{12}+x_{5}x_{13}+x_{6}x_{14}+x_{7}x_{15}+x_{8}x_{16})}{c}}}
u
3
=
(
x
1
2
+
x
2
2
+
a
x
3
2
+
x
4
2
+
x
5
2
+
x
6
2
+
x
7
2
+
x
8
2
)
x
11
−
2
x
3
(
x
1
x
9
+
x
2
x
10
+
b
x
3
x
11
+
x
4
x
12
+
x
5
x
13
+
x
6
x
14
+
x
7
x
15
+
x
8
x
16
)
c
{\displaystyle u_{3}={\tfrac {(x_{1}^{2}+x_{2}^{2}+ax_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2})x_{11}-2x_{3}(x_{1}x_{9}+x_{2}x_{10}+bx_{3}x_{11}+x_{4}x_{12}+x_{5}x_{13}+x_{6}x_{14}+x_{7}x_{15}+x_{8}x_{16})}{c}}}
u
4
=
(
x
1
2
+
x
2
2
+
x
3
2
+
a
x
4
2
+
x
5
2
+
x
6
2
+
x
7
2
+
x
8
2
)
x
12
−
2
x
4
(
x
1
x
9
+
x
2
x
10
+
x
3
x
11
+
b
x
4
x
12
+
x
5
x
13
+
x
6
x
14
+
x
7
x
15
+
x
8
x
16
)
c
{\displaystyle u_{4}={\tfrac {(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+ax_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2})x_{12}-2x_{4}(x_{1}x_{9}+x_{2}x_{10}+x_{3}x_{11}+bx_{4}x_{12}+x_{5}x_{13}+x_{6}x_{14}+x_{7}x_{15}+x_{8}x_{16})}{c}}}
u
5
=
(
x
1
2
+
x
2
2
+
x
3
2
+
x
4
2
+
a
x
5
2
+
x
6
2
+
x
7
2
+
x
8
2
)
x
13
−
2
x
5
(
x
1
x
9
+
x
2
x
10
+
x
3
x
11
+
x
4
x
12
+
b
x
5
x
13
+
x
6
x
14
+
x
7
x
15
+
x
8
x
16
)
c
{\displaystyle u_{5}={\tfrac {(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+ax_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2})x_{13}-2x_{5}(x_{1}x_{9}+x_{2}x_{10}+x_{3}x_{11}+x_{4}x_{12}+bx_{5}x_{13}+x_{6}x_{14}+x_{7}x_{15}+x_{8}x_{16})}{c}}}
u
6
=
(
x
1
2
+
x
2
2
+
x
3
2
+
x
4
2
+
x
5
2
+
a
x
6
2
+
x
7
2
+
x
8
2
)
x
14
−
2
x
6
(
x
1
x
9
+
x
2
x
10
+
x
3
x
11
+
x
4
x
12
+
x
5
x
13
+
b
x
6
x
14
+
x
7
x
15
+
x
8
x
16
)
c
{\displaystyle u_{6}={\tfrac {(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+ax_{6}^{2}+x_{7}^{2}+x_{8}^{2})x_{14}-2x_{6}(x_{1}x_{9}+x_{2}x_{10}+x_{3}x_{11}+x_{4}x_{12}+x_{5}x_{13}+bx_{6}x_{14}+x_{7}x_{15}+x_{8}x_{16})}{c}}}
u
7
=
(
x
1
2
+
x
2
2
+
x
3
2
+
x
4
2
+
x
5
2
+
x
6
2
+
a
x
7
2
+
x
8
2
)
x
15
−
2
x
7
(
x
1
x
9
+
x
2
x
10
+
x
3
x
11
+
x
4
x
12
+
x
5
x
13
+
x
6
x
14
+
b
x
7
x
15
+
x
8
x
16
)
c
{\displaystyle u_{7}={\tfrac {(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+ax_{7}^{2}+x_{8}^{2})x_{15}-2x_{7}(x_{1}x_{9}+x_{2}x_{10}+x_{3}x_{11}+x_{4}x_{12}+x_{5}x_{13}+x_{6}x_{14}+bx_{7}x_{15}+x_{8}x_{16})}{c}}}
u
8
=
(
x
1
2
+
x
2
2
+
x
3
2
+
x
4
2
+
x
5
2
+
x
6
2
+
x
7
2
+
a
x
8
2
)
x
16
−
2
x
8
(
x
1
x
9
+
x
2
x
10
+
x
3
x
11
+
x
4
x
12
+
x
5
x
13
+
x
6
x
14
+
x
7
x
15
+
b
x
8
x
16
)
c
{\displaystyle u_{8}={\tfrac {(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+ax_{8}^{2})x_{16}-2x_{8}(x_{1}x_{9}+x_{2}x_{10}+x_{3}x_{11}+x_{4}x_{12}+x_{5}x_{13}+x_{6}x_{14}+x_{7}x_{15}+bx_{8}x_{16})}{c}}}
and,
a
=
−
1
,
b
=
0
,
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
.
{\displaystyle a=-1,\;\;b=0,\;\;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}\,.}
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
{\displaystyle u_{i}}
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
9
2
+
x
10
2
+
x
11
2
+
x
12
2
+
x
13
2
+
x
14
2
+
x
15
2
+
x
16
2
{\displaystyle 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_{9}^{2}+x_{10}^{2}+x_{11}^{2}+x_{12}^{2}+x_{13}^{2}+x_{14}^{2}+x_{15}^{2}+x_{16}^{2}}
No sixteen-square identity exists involving only bilinear functions since Hurwitz's theorem states an identity of the form
(
x
1
2
+
x
2
2
+
x
3
2
+
⋯
+
x
n
2
)
(
y
1
2
+
y
2
2
+
y
3
2
+
⋯
+
y
n
2
)
=
z
1
2
+
z
2
2
+
z
3
2
+
⋯
+
z
n
2
{\displaystyle (x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+\cdots +x_{n}^{2})(y_{1}^{2}+y_{2}^{2}+y_{3}^{2}+\cdots +y_{n}^{2})=z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+\cdots +z_{n}^{2}}
with the
z
i
{\displaystyle z_{i}}
bilinear functions of the
x
i
{\displaystyle x_{i}}
y
i
{\displaystyle y_{i}}
n ∈ {1, 2, 4, 8} . However, the more general (1965) shows that if the
z
i
{\displaystyle z_{i}}
rational functions of one set of variables, hence has a denominator , then it is possible for all
n
=
2
m
{\displaystyle n=2^{m}}
[3] Euler's four-square and Degen's eight-square identities.
See also [ ] References [ ]
^ 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
^ A. Pfister, Zur Darstellung von -1 als Summe von Quadraten in einem Körper," J. London Math. Soc. 40 (1965), 159-165
^ Pfister's Theorem on Sums of Squares, Keith Conrad, http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/pfister.pdf
External links [ ] 
