When we look at the two smallest numbers $n$ such that $n$ can not be written as the sum of $3$ squares, we get $7$.
And we know that there exists a vector product on $\mathbb R^n$ if, and only if $n\in\{2,3,7\}$.
Could there be a connection? $3$ and $7$ both appear here...
Maybe it is linked to the fact that vector product has to do with quaternions, and so does Lagrange's theorem on the sum of four squares.
Additional informations here.
