Euler's four-square identity explained

In mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four squares, is itself a sum of four squares.

Algebraic identity

For any pair of quadruples from a commutative ring, the following expressions are equal:

$\begin& \left(a_1^2+a_2^2+a_3^2+a_4^2\right) \left(b_1^2+b_2^2+b_3^2+b_4^2\right) \\[3mu]&\qquad = \left(a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4\right)^2 + \left(a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3\right)^2 \\[3mu]
&\qquad\qquad+ \left(a_1 b_3 - a_2 b_4 + a_3 b_1 + a_4 b_2\right)^2+ \left(a_1 b_4 + a_2 b_3 - a_3 b_2 + a_4 b_1\right)^2.\end$

Euler wrote about this identity in a letter dated May 4, 1748 to Goldbach^[1] ^[2] (but he used a different sign convention from the above). It can be verified with elementary algebra.

The identity was used by Lagrange to prove his 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

a_k

-a_k

, and/or any

b_k

-b_k

If the

a_k

and

b_k

are real numbers, the identity expresses the fact that the absolute value of the product of two quaternions is equal to the product of their absolute values, in the same way that the Brahmagupta–Fibonacci two-square identity does for complex numbers. This property is the definitive feature of composition algebras.

Hurwitz's theorem states that an identity of form,

$\left(a_1^2+a_2^2+a_3^2+\dots+a_n^2\right)\left(b_1^2+b_2^2+b_3^2+\dots+b_n^2\right) = c_1^2+c_2^2+c_3^2+ \dots + c_n^2$

where the

c_i

are bilinear functions of the

a_i

and

b_i

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: . This defines the quaternion multiplication rule, 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

\alpha=a₁+a₂i+a₃j+a₄k

and

\beta=b₁+b₂i+b₃j+b₄k

be a pair of quaternions. Their quaternion conjugates are

\alpha^*=a₁-a₂i-a₃j-a₄k

and

\beta^*=b₁-b₂i-b₃j-b₄k

. Then

$A := \alpha \alpha^* = a_1^2 + a_2^2 + a_3^2 + a_4^2$

and

$B := \beta \beta^* = b_1^2 + b_2^2 + b_3^2 + b_4^2.$

The product of these two is

AB=\alpha\alpha^*\beta\beta^*

, where

\beta\beta^*

is a real number, so it can commute with the quaternion

\alpha^*

, yielding

$A B = \alpha \beta \beta^* \alpha^*.$

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

$A B = \alpha \beta (\alpha \beta)^* = \gamma \gamma^*$

where

\gamma

is the Hamilton product of

\alpha

and

\beta

$\begin\gamma &= \left(a_1 + \langle a_2, a_3, a_4 \rangle\right) \left(b_1 + \langle b_2, b_3, b_4 \rangle\right) \\[3mu]& = a_1 b_1 + a_1 \langle b_2, \ b_3, \ b_4\rangle + \langle a_2, \ a_3, \ a_4\rangle b_1 + \langle a_2, \ a_3, \ a_4\rangle \langle b_2, \ b_3, \ b_4\rangle \\[3mu]& = a_1 b_1 + \langle a_1 b_2, \ a_1 b_3, \ a_1 b_4\rangle + \langle a_2 b_1, \ a_3 b_1, \ a_4 b_1\rangle \\&\qquad - \langle a_2,\ a_3, \ a_4\rangle \cdot \langle b_2, \ b_3, \ b_4\rangle + \langle a_2, \ a_3, \ a_4\rangle \times \langle b_2, \ b_3, \ b_4\rangle \\[3mu]& = a_1 b_1 + \langle a_1 b_2 + a_2 b_1, \ a_1 b_3 + a_3 b_1, \ a_1 b_4 + a_4 b_1\rangle \\&\qquad - a_2 b_2 - a_3 b_3 - a_4 b_4 + \langle a_3 b_4 - a_4 b_3, \ a_4 b_2 - a_2 b_4, \ a_2 b_3 - a_3 b_2\rangle \\[3mu]& = (a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4) \\&\qquad + \langle a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3, \ a_1 b_3 + a_3 b_1 + a_4 b_2 - a_2 b_4, \ a_1 b_4 + a_4 b_1 + a_2 b_3 - a_3 b_2\rangle \\[3mu]\gamma &= (a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4) + (a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3) i \\&\qquad + (a_1 b_3 + a_3 b_1 + a_4 b_2 - a_2 b_4) j + (a_1 b_4 + a_4 b_1 + a_2 b_3 - a_3 b_2) k.\end$

Then

$\begin\gamma^* &= (a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4) - (a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3) i \\&\qquad - (a_1 b_3 + a_3 b_1 + a_4 b_2 - a_2 b_4) j - (a_1 b_4 + a_4 b_1 + a_2 b_3 - a_3 b_2) k .\end$

\gamma=r+\vecu

where

is the scalar part and

\vecu=\langleu_1,u_2,u_3\rangle

is the vector part, then

\gamma^*=r-\vecu

$\begin\gamma \gamma^* &= (r + \vec u) (r - \vec u) = r^2 - r \vec u + r \vec u - \vec u \vec u = r^2 + \vec u \cdot \vec u - \vec u \times \vec u \\&= r^2 + \vec u \cdot \vec u = r^2 + u_1^2 + u_2^2 + u_3^2.\end$

So,

$\beginA B = \gamma \gamma^* &= (a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4)^2 + (a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3)^2 \\&\qquad + (a_1 b_3 + a_3 b_1 + a_4 b_2 - a_2 b_4)^2 + (a_1 b_4 + a_4 b_1 + a_2 b_3 - a_3 b_2)^2.\end$

Pfister's identity

Pfister found another square identity for any even power:^[3]

If the

c_i

are just rational functions of one set of variables, so that each

c_i

has a denominator, then it is possible for all

n=2^m

Thus, another four-square identity is as follows: $\begin&\left(a_1^2+a_2^2+a_3^2+a_4^2\right)\left(b_1^2+b_2^2+b_3^2+b_4^2\right) \\[5mu]
&\quad= \left(a_1 b_4 + a_2 b_3 + a_3 b_2 + a_4 b_1\right)^2 + \left(a_1 b_3 - a_2 b_4 + a_3 b_1 - a_4 b_2\right)^2 \\
&\quad\qquad+ \left(a_1 b_2 + a_2 b_1 + \frac - \frac\right)^2
+ \left(a_1 b_1 - a_2 b_2 - \frac - \frac\right)^2\end$

where

u₁

and

u₂

are given by

\beginu_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

Incidentally, the following identity is also true:

$u_1^2+u_2^2 = \left(b_1^2+b_2^2\right)^2\left(b_3^2+b_4^2\right)$

References

Leonhard Euler: Life, Work and Legacy, R.E. Bradley and C.E. Sandifer (eds), Elsevier, 2007, p. 193
Mathematical Evolutions, A. Shenitzer and J. Stillwell (eds), Math. Assoc. America, 2002, p. 174
Keith Conrad Pfister's Theorem on Sums of Squares from University of Connecticut

External links

A Collection of Algebraic Identities
http://math.dartmouth.edu/~euler/correspondence/letters/OO0841.pdf Lettre CXV from Euler to Goldbach

Euler's four-square identity explained

Algebraic identity

Proof of the identity using quaternions

Pfister's identity

See also

References

External links