mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...

, 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)
=  &

\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 \\

&+ \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 Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

wrote about this identity in a letter dated May 4, 1748 to Goldbach (but he used a different sign convention from the above). It can be verified with elementary algebra. The identity was used by

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiafour square theorem. More specifically, it implies that it is sufficient to prove the theorem for

prime numbers A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

, 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

quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...

s 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 = a_1 + a_2 i + a_3 j + a_4 k

and

\beta = b_1 + b_2 i + b_3 j + b_4 k

be a pair of quaternions. Their quaternion conjugates are

\alpha^* = a_1 - a_2 i - a_3 j - a_4 k

and

\beta^* = b_1 - b_2 i - b_3 j - b_4 k

. 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

A B = \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 In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...

\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) \\
& = 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 \\
& = 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 - \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 \\
& = 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 - 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 \\
& = (a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4) + \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 \\
\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 + (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

\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 - (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 .

\gamma = r + \vec u

where

r

is the scalar part and

\vec u = \langle u_1, u_2, u_3\rangle

is the vector part, then

\gamma^* = r - \vec u

\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.

So,

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 + (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.

Pfister's identity

Pfister found another square identity for any even power:Keith Conra
Pfister's Theorem on Sums of Squares
from University of Connecticut 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) \\

&= \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 \\

&\;\;+ \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

u_1

and

u_2

are given by

\begin
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

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

External links

A Collection of Algebraic Identities
Lettre CXV from Euler to Goldbach {{DEFAULTSORT:Euler's Four-Square Identity Elementary algebra Elementary number theory Mathematical identities Squares in number theory Leonhard Euler

Algebraic identity

Proof of the identity using quaternions

Pfister's identity

See also

References

External links