Euler's four-square identity

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 identityEdit

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

 

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   to  , and/or any   to  .

If the   and   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,

 

where the   are bilinear functions of the   and   is possible only for n = 1, 2, 4, or 8.

Proof of the identity using quaternionsEdit

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: (a·a)(b·b) = (a×b)·(a×b). This defines the quaternion multiplication rule 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   and   be a pair of quaternions. Their quaternion conjugates are   and  . Then

 
and
 

The product of these two is  , where   is a real number, so it can commute with the quaternion  , yielding

 
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
 
where   is the Hamilton product of   and  :
 
Then
 
If   where   is the scalar part and   is the vector part, then   so
 
So,
 

Pfister's identityEdit

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

If the   are just rational functions of one set of variables, so that each   has a denominator, then it is possible for all  .

Thus, another four-square identity is as follows:

 

where   and   are given by

 

Incidentally, the following identity is also true:

 

See alsoEdit

ReferencesEdit

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

External linksEdit