3
$\begingroup$

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.

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    I expect there to be no connection. The connection between normed division algebras, (multiplicative) sums of square identities and binary cross products that *does* exist occurs when the division algebra has dimension $2^n$, the sums are of $2^n$ squares, and the cross product is on a real inner product space of dimension $2^n-1$, where $n=0,1,2,3$. [Did you accidentally link to the wrong thing BTW?] – 2017-01-21
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    Also, your list of $n$ for which binary cross products exist on $\Bbb R^n$ is incorrect, it should be $0,1,3,7$ (the first two being degenerate). There are also other cross products - multiplication by $i$ on $\Bbb C^n$ is a unary one if we interpret it as $\Bbb R^{2n}$, there is always two counary cross products corresponding to the two orientations of space, and an exotic ternary cross product in eight-dimensions. – 2017-01-21

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