Euler's four-square identity

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

(a_1^2%2Ba_2^2%2Ba_3^2%2Ba_4^2)(b_1^2%2Bb_2^2%2Bb_3^2%2Bb_4^2)=\,
(a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4)^2 %2B\,
(a_1 b_2 %2B a_2 b_1 %2B a_3 b_4 - a_4 b_3)^2 %2B\,
(a_1 b_3 - a_2 b_4 %2B a_3 b_1 %2B a_4 b_2)^2 %2B\,
(a_1 b_4 %2B a_2 b_3 - a_3 b_2 %2B a_4 b_1)^2.\,

Euler wrote about this identity in a letter dated May 4, 1748 to Goldbach[1][2] (but note that he used a different sign convention from the above). It can be proven with elementary algebra and holds in every commutative ring. If the a_k and b_k are real numbers, a more elegant proof is available: 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.

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 to -a_k, b_k to -b_k, or by changing the signs inside any of the squared terms on the right hand side.

However, a different kind of four-square identity can be given as,

(a_1^2%2Ba_2^2%2Ba_3^2%2Ba_4^2)(b_1^2%2Bb_2^2%2Bb_3^2%2Bb_4^2)=\,
(a_1 b_4 %2B a_2 b_3 %2B a_3 b_2 %2B a_4 b_1)^2 %2B\,
(a_1 b_3 - a_2 b_4 %2B a_3 b_1 - a_4 b_2)^2 %2B\,
\left(a_1 b_2 %2B a_2 b_1 %2B \frac{a_3 (b_1^2b_4-2b_1b_2b_3-b_2^2b_4)}{b_1^2%2Bb_2^2} - \frac{a_4 (b_1^2b_3%2B2b_1b_2b_4-b_2^2b_3)}{b_1^2%2Bb_2^2}\right)^2%2B\,
\left(a_1 b_1 - a_2 b_2 - \frac{a_4 (b_1^2b_4-2b_1b_2b_3-b_2^2b_4)}{b_1^2%2Bb_2^2} - \frac{a_3 (b_1^2b_3%2B2b_1b_2b_4-b_2^2b_3)}{b_1^2%2Bb_2^2}\right)^2.\,

More generally, while Hurwitz's theorem states that an identity of form,

(a_1^2%2Ba_2^2%2Ba_3^2%2B...%2Ba_n^2)(b_1^2%2Bb_2^2%2Bb_3^2%2B...%2Bb_n^2) = c_1^2%2Bc_2^2%2Bc_3^2%2B...%2Bc_n^2\,

where the c_i are bilinear functions of the a_i and b_i is possible only for n = {1, 2, 4, 8}, the more general Pfister's theorem allows that if the c_i are just rational functions of one set of variables (in this case, the b_i), hence has a denominator, then it is possible for all n = 2^m.[3]

Euler's Four Square Identity can be further Generalized by letting

A,B, and C be the Coefficients of four Binary Quartic Forms, whose Variables are ( ~ u_0, ~ v_0 ~ ), ~ ( ~ u_1, ~ v_1 ~ ),~ ( ~ x_0, ~y_0 ~ ) , and ( ~ x_1, ~ y_1 ~ )
Also let \alpha, ~ \beta, \mu, and, \xi be any quantities desired. Then
The derivation of this is done in pure algebra. The algebra needed to derive this Identity can be found in a PDF file called the Generalization of Euler's Four Squares (about 60 pages ). It can be found at http://www.samz.us

Notice that we have Euler's Identity by letting A ~ = ~ 1,~ B ~ =~ 0 , and C ~ = ~ 1 , and \alpha,~ \beta, ~ \mu , and, \xi ~ = ~ 1

See also

References

  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. ^ Pfister's Theorem on Sums of Squares, Keith Conrad, http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/pfister.pdf

External links