Editing Euler's four-square identity

{{short description|Product of sums of four squares expressed as a sum of four squares}}
In [[mathematics]], '''Euler's four-square identity''' says that the product of two numbers, each of which is a sum of four [[square (algebra)|square]]s, is itself a sum of four squares.

==Algebraic identity==
For any pair of quadruples from a [[commutative ring]], the following expressions are equal:

<math display="block">\begin{align}
& \left(a_1^2+a_2^2+a_3^2+a_4^2\right) \left(b_1^2+b_2^2+b_3^2+b_4^2\right) \\[3mu]
&\qquad = \left(a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4\right)^2 
+ \left(a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3\right)^2 \\[3mu]

&\qquad\qquad+ \left(a_1 b_3 - a_2 b_4 + a_3 b_1 + a_4 b_2\right)^2
+ \left(a_1 b_4 + a_2 b_3 - a_3 b_2 + a_4 b_1\right)^2.
\end{align}</math>

[[Leonhard Euler|Euler]] wrote about this identity in a letter dated May 4, 1748 to [[Christian Goldbach|Goldbach]]<ref>''Leonhard Euler: Life, Work and Legacy'', R.E. Bradley and C.E. Sandifer (eds), Elsevier, 2007, p. 193</ref><ref>''Mathematical Evolutions'', A. Shenitzer and J. Stillwell (eds), Math. Assoc. America, 2002, p. 174</ref> (but he used a different sign convention from the above). It can be verified with [[elementary algebra]]. 

The identity was used by [[Joseph Louis Lagrange|Lagrange]] to prove his [[Lagrange's four-square theorem|four square theorem]]. More specifically, it implies that it is sufficient to prove the theorem for [[prime numbers]], after which the more general theorem follows. The sign convention used above corresponds to the signs obtained by multiplying two quaternions. Other sign conventions can be obtained by changing any <math>a_k</math> to <math>-a_k</math>, and/or any <math>b_k</math> to <math>-b_k</math>.

If the <math>a_k</math> and <math>b_k</math> are [[real number]]s, the identity expresses the fact that the absolute value of the product of two [[quaternion]]s is equal to the product of their absolute values, in the same way that the [[Brahmagupta–Fibonacci identity|Brahmagupta–Fibonacci two-square identity]] does for [[complex numbers]]. This property is the definitive feature of [[composition algebra]]s.

[[Hurwitz's theorem (normed division algebras)|Hurwitz's theorem]] states that an identity of form,

<math display="block">\left(a_1^2+a_2^2+a_3^2+\dots+a_n^2\right)\left(b_1^2+b_2^2+b_3^2+\dots+b_n^2\right) = c_1^2+c_2^2+c_3^2+ \dots + c_n^2</math>

where the <math>c_i</math> are [[bilinear map|bilinear]] functions of the <math>a_i</math> and <math>b_i</math> is possible only for ''n'' = 1, 2, 4, or 8.

===Proof of the identity using quaternions===
Comment: The proof of Euler's four-square identity is by simple algebraic evaluation. Quaternions derive from the four-square identity, which can be written as the product of two inner products of 4-dimensional vectors, yielding again an inner product of 4-dimensional vectors: {{math|1=(''a''·''a'')(''b''·''b'') = (''a''×''b'')·(''a''×''b'')}}. This defines the quaternion multiplication rule {{math|''a''×''b''}}, which simply reflects Euler's identity, and some mathematics of quaternions. Quaternions are, so to say, the "square root" of the four-square identity. But let the proof go on: 

Let <math>\alpha = a_1 + a_2 i + a_3 j + a_4 k</math> and <math>\beta = b_1 + b_2 i + b_3 j + b_4 k</math> be a pair of quaternions. Their quaternion conjugates are <math>\alpha^* = a_1 - a_2 i - a_3 j - a_4 k </math> and <math>\beta^* = b_1 - b_2 i - b_3 j - b_4 k</math>. Then

<math display="block">A := \alpha \alpha^* = a_1^2 + a_2^2 + a_3^2 + a_4^2</math>

and

<math display="block">B := \beta \beta^* = b_1^2 + b_2^2 + b_3^2 + b_4^2.</math>

The product of these two is <math>A B = \alpha \alpha^* \beta \beta^*</math>, where <math>\beta \beta^*</math> is a real number, so it can commute with the quaternion <math>\alpha^*</math>, yielding

<math display="block">A B = \alpha \beta \beta^* \alpha^*.</math>

No parentheses are necessary above, because quaternions [[Associative property|associate]]. The conjugate of a product is equal to the commuted product of the conjugates of the product's factors, so

<math display="block">A B = \alpha \beta (\alpha \beta)^* = \gamma \gamma^*</math>

where <math>\gamma</math> is the [[Quaternion#Hamilton product|Hamilton product]] of <math>\alpha</math> and <math>\beta</math>:

<math display="block">\begin{align}
\gamma &= \left( a_1 + \langle a_2, a_3, a_4 \rangle\right) \left(b_1 + \langle b_2, b_3, b_4 \rangle\right) \\[3mu]
& = a_1 b_1 + a_1 \langle b_2, \ b_3, \ b_4\rangle + \langle a_2, \ a_3, \ a_4\rangle b_1 + \langle a_2, \ a_3, \ a_4\rangle \langle b_2, \ b_3, \ b_4\rangle \\[3mu]
& = a_1 b_1 + \langle a_1 b_2, \ a_1 b_3, \ a_1 b_4\rangle + \langle a_2 b_1, \ a_3 b_1, \ a_4 b_1\rangle \\
&\qquad - \langle a_2,\  a_3, \ a_4\rangle \cdot \langle b_2, \ b_3, \ b_4\rangle + \langle a_2, \ a_3, \ a_4\rangle \times \langle b_2, \ b_3, \  b_4\rangle \\[3mu]
& = a_1 b_1 + \langle a_1 b_2 + a_2 b_1, \ a_1 b_3 + a_3 b_1, \ a_1 b_4 + a_4 b_1\rangle \\
&\qquad - a_2 b_2 - a_3 b_3 - a_4 b_4 + \langle a_3 b_4 - a_4 b_3, \ a_4 b_2 - a_2 b_4, \ a_2 b_3 - a_3 b_2\rangle \\[3mu]
& = (a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4) \\
&\qquad + \langle a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3, \ a_1 b_3 + a_3 b_1 + a_4 b_2 - a_2 b_4, \ a_1 b_4 + a_4 b_1 + a_2 b_3 - a_3 b_2\rangle \\[3mu]
\gamma &= (a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4) + (a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3) i \\
&\qquad + (a_1 b_3 + a_3 b_1 + a_4 b_2 - a_2 b_4) j + (a_1 b_4 + a_4 b_1 + a_2 b_3 - a_3 b_2) k.
\end{align}</math>

Then

<math display="block">\begin{align}
\gamma^* &= (a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4) - (a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3) i \\
&\qquad - (a_1 b_3 + a_3 b_1 + a_4 b_2 - a_2 b_4) j - (a_1 b_4 + a_4 b_1 + a_2 b_3 - a_3 b_2) k .
\end{align}</math>

If <math>\gamma = r + \vec u</math> where <math>r</math> is the scalar part and <math>\vec u = \langle u_1, u_2, u_3\rangle</math> is the vector part, then <math>\gamma^* = r - \vec u</math> so

<math display="block">\begin{align}
\gamma \gamma^* &= (r + \vec u) (r - \vec u) = r^2 - r \vec u + r \vec u - \vec u \vec u = r^2 + \vec u \cdot \vec u - \vec u \times \vec u \\
&= r^2 + \vec u \cdot \vec u = r^2 + u_1^2 + u_2^2 + u_3^2.
\end{align}</math>

So,

<math display="block">\begin{align}
A B = \gamma \gamma^* &= (a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4)^2 + (a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3)^2 \\
&\qquad + (a_1 b_3 + a_3 b_1 + a_4 b_2 - a_2 b_4)^2 + (a_1 b_4 + a_4 b_1 + a_2 b_3 - a_3 b_2)^2.
\end{align}</math>

==Pfister's identity==
Pfister found another square identity for any even power:<ref>Keith Conrad [http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/pfister.pdf Pfister's Theorem on Sums of Squares] from [[University of Connecticut]]</ref> 

If the <math>c_i</math> are just [[rational functions]] of one set of variables, so that each <math>c_i</math> has a [[denominator]], then it is possible for all <math>n = 2^m</math>.

Thus, another four-square identity is as follows:
<math display="block">\begin{align}
&\left(a_1^2+a_2^2+a_3^2+a_4^2\right)\left(b_1^2+b_2^2+b_3^2+b_4^2\right) \\[5mu]

&\quad= \left(a_1 b_4 + a_2 b_3 + a_3 b_2 + a_4 b_1\right)^2 + \left(a_1 b_3 - a_2 b_4 + a_3 b_1 - a_4 b_2\right)^2 \\

&\quad\qquad+ \left(a_1 b_2 + a_2 b_1 + \frac{a_3 u_1}{b_1^2+b_2^2} - \frac{a_4 u_2}{b_1^2+b_2^2}\right)^2

+ \left(a_1 b_1 - a_2 b_2 - \frac{a_4 u_1}{b_1^2+b_2^2} - \frac{a_3 u_2}{b_1^2+b_2^2}\right)^2
\end{align}</math>

where <math>u_1</math> and <math>u_2</math> are given by 
<math display="block">\begin{align}
u_1 &= b_1^2 b_4 - 2 b_1 b_2 b_3 - b_2^2 b_4 \\
u_2 &= b_1^2 b_3 + 2 b_1 b_2 b_4 - b_2^2 b_3
\end{align}</math>

Incidentally, the following identity is also true: 

<math display="block">u_1^2+u_2^2 = \left(b_1^2+b_2^2\right)^2\left(b_3^2+b_4^2\right)</math>

==See also==
*[[Brahmagupta–Fibonacci identity]] (sums of two squares)
*[[Degen's eight-square identity]]
*[[Pfister's sixteen-square identity]]
*[[Latin square]]

==References==
<references/>

==External links==
*[http://sites.google.com/site/tpiezas/005b/ A Collection of Algebraic Identities] {{Webarchive|url=https://web.archive.org/web/20120306122543/http://sites.google.com/site/tpiezas/005b |date=2012-03-06 }}
*[http://math.dartmouth.edu/~euler/correspondence/letters/OO0841.pdf] Lettre CXV from Euler to Goldbach

{{Leonhard Euler}}
{{DEFAULTSORT:Euler's Four-Square Identity}}
[[Category:Algebraic identities]]
[[Category:Elementary number theory]]
[[Category:Squares in number theory]]
[[Category:Leonhard Euler]]
[[Category:Quaternions]]