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Denote by $ K(d) $ the kissing number in dimension $ d $.

I have two questions : 1) does $ d\mid K(d) $ for all $ d $? 2) does $ d\mid D $ imply $K(d)\mid K(D) $?

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    $\begingroup$ I think this question will be very difficult to answer. We do not currently have enough "data" on $K(n)$ for $n\neq 1,2,3,4,8,24$, so we can't numerically ascertain either. $\endgroup$ – YiFan Jan 31 at 21:15
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As already noted in the comments, these are open questions. However, the answer cannot be "yes" to both.

If the answer to (1) is "yes", then since it is known $40\leq K(5)\leq 44$, we can conclude $K(5)=40$. It is also known that $500\leq K(10)\leq 554$, so we'd conclude $K(10)\in\{500,510,520,530,540,550\}$.

If the answer to (2) is "yes", we have $5\mid 10$, so we'd need $K(10)=520$. Yet also $2\mid 10$, and $K(2)=6\nmid 520$.

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