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 identity

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

$(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2})(b_{1}^{2}+b_{2}^{2}+b_{3}^{2}+b_{4}^{2})=\,$

$(a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}+a_{4}b_{4})^{2}+\,$

$(a_{1}b_{2}-a_{2}b_{1}+a_{3}b_{4}-a_{4}b_{3})^{2}+\,$

$(a_{1}b_{3}-a_{2}b_{4}-a_{3}b_{1}+a_{4}b_{2})^{2}+\,$

$(a_{1}b_{4}+a_{2}b_{3}-a_{3}b_{2}-a_{4}b_{1})^{2}.\,$

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 proven 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 $a_{k}$ to $-a_{k}$, and/or any $b_{k}$ to $-b_{k}$.

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,

$(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+...+a_{n}^{2})(b_{1}^{2}+b_{2}^{2}+b_{3}^{2}+...+b_{n}^{2})=c_{1}^{2}+c_{2}^{2}+c_{3}^{2}+...+c_{n}^{2}\,$

where the $c_{i}$ are bilinear functions of the $a_{i}$ and $b_{i}$ is possible only for n = 1, 2, 4, or 8.

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, hence has a denominator, then it is possible for all $n=2^{m}$.

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

$(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2})(b_{1}^{2}+b_{2}^{2}+b_{3}^{2}+b_{4}^{2})=\,$

$(a_{1}b_{4}+a_{2}b_{3}+a_{3}b_{2}+a_{4}b_{1})^{2}+\,$

$(a_{1}b_{3}-a_{2}b_{4}+a_{3}b_{1}-a_{4}b_{2})^{2}+\,$

$\left(a_{1}b_{2}+a_{2}b_{1}+{\frac {a_{3}u_{1}}{b_{1}^{2}+b_{2}^{2}}}-{\frac {a_{4}u_{2}}{b_{1}^{2}+b_{2}^{2}}}\right)^{2}+\,$

$\left(a_{1}b_{1}-a_{2}b_{2}-{\frac {a_{4}u_{1}}{b_{1}^{2}+b_{2}^{2}}}-{\frac {a_{3}u_{2}}{b_{1}^{2}+b_{2}^{2}}}\right)^{2}\,$

where,

$u_{1}=b_{1}^{2}b_{4}-2b_{1}b_{2}b_{3}-b_{2}^{2}b_{4}$

$u_{2}=b_{1}^{2}b_{3}+2b_{1}b_{2}b_{4}-b_{2}^{2}b_{3}$

Note also the incidental fact that,

$u_{1}^{2}+u_{2}^{2}=(b_{1}^{2}+b_{2}^{2})^{2}(b_{3}^{2}+b_{4}^{2})$

See also

• Brahmagupta–Fibonacci identity (sums of two squares)
• Degen's eight-square identity
• Pfister's sixteen-square identity
• Latin square

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. Keith Conrad Pfister's Theorem on Sums of Squares from University of Connecticut

External links

• A Collection of Algebraic Identities
• Lettre CXV from Euler to Goldbach