Euler Four-Square Identity

The amazing polynomial identity

$({a_1}^2+{a_2}^2+{a_3}^2+{a_4}^2)({b_1}^2+{b_2}^2+{b_3}^2+{b_4}^2)$

$\quad =(a_1b_1-a_2b_2-a_3b_3-a_4b_4)^2+(a_1b_2+a_2b_1+a_3b_4-a_4b_3)^2$

$\quad +(a_1b_3-a_2b_4+a_3b_1+a_4b_2)^2+(a_1b_4+a_2b_3-a_3b_2+a_4b_1)^2,$

communicated by Euler in a letter to Goldbach on April 15, 1705. The identity also follows from the fact that the norm of the product of two Quaternions is the product of the norms (Conway and Guy 1996).

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 232, 1996.

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A. K. Peters, p. 8, 1996.

$({a_1}^2+{a_2}^2+{a_3}^2+{a_4}^2)({b_1}^2+{b_2}^2+{b_3}^2+{b_4}^2)$
$\quad =(a_1b_1-a_2b_2-a_3b_3-a_4b_4)^2+(a_1b_2+a_2b_1+a_3b_4-a_4b_3)^2$
$\quad +(a_1b_3-a_2b_4+a_3b_1+a_4b_2)^2+(a_1b_4+a_2b_3-a_3b_2+a_4b_1)^2,$