It can be proved easily using complex numbers that the product of sums of two squares is a sum of two squares:
Euler noted in 1743 that the product of sums of four squares is a sum of four squares:
This fact can be proved using quaternions, a generalization of complex numbers to four dimensions. A corresponding result holds for sums of eight squares. It is worth mentioning that no similar results exist for 16 squares or any number of squares other than 1, 2, 4 and 8.
