Let $k$ be a positive integer and let $p$ be a prime. In my 2011 PAMS paper joint with my former student W. Zhang [Proc. Amer. Math. Soc. 139(2011), 1569-1577], we studied when $$S(k)=\left\{\binom nk:\ n=0,1,2,\ldots\right\}$$ is a dense subset of the ring $\mathbb Z_p$ of $p$-padic integers (i.e., $S(k)$ contains a complete system of residues modulo any powers of $p$).

We call a positive integer $k$ *universal* if $S(k)$ is dense in $\mathbb Z_p$ for any prime $p\le k$. I and W. Zhang showed that
$$1,\ 2,\ 3,\ 4,\ 5,\ 9,\ 11,\ 17,\ 29\tag{1}$$
are universal. We believe that there are no other universal numbers.

**Question.** Can one show that there is no universal number not listed in $(1)$?

Your comments are welcome!