Tuesday, 7 February 2012

Products of sums of squares

I decided to do some math today.  Some interesting results about products of sums of (integer) squares ...

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.

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