# Divisibility of kissing numbers

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 $$K(d)\mid K(D)$$?

• 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. – YiFan Jan 31 at 21:15

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 $$K(2)=6\nmid 520$$.