Ramanujan was fascinated by algebraic identities and gave many beautiful examples, like the "Ramanujan 6-10-8 Identity". In this article, we discuss the product of two sums of n squares equal to the sum of n squares and give the explicit identities for n = 1, 2, 4, 8.
The importance of these is that they are intimately connected to certain division algebras, namely for the reals, complex numbers, quaternions, and octonions. They also automatically lead to Diophantine identities of squares equal to n+1 squares and certain magic squares.
However, if we relax some conditions, we can actually find a parametric identity even for n = 16 or, in general, for n a power of 2 and we will give a numerical example.
