Euler's four-square identity

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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[edit]

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 quaternions[edit]

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


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 :


If where is the scalar part and is the vector part, then so


Pfister's identity[edit]

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 also[edit]


  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 links[edit]