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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!

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    $\begingroup$ Could you link to your paper (using the AMS site version, not the paywall JSTOR version)? $\endgroup$ – YCor Oct 15 at 19:18
  • $\begingroup$ I have added the link. Thank you for the suggestion. $\endgroup$ – Zhi-Wei Sun Oct 15 at 19:25
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    $\begingroup$ @Zhi-Wei Sun: Could you, perhaps, summarize what the obstacles were in obstructing your approach in Theorem 1.2 of the linked paper from resolving the problem. Was it a matter of computation (as probably hinted by your statement right after Theorem 1.3; perhaps a formulation of a “combinatorial” conjecture —arising from Theorem 1.2–might be equivalent (or otherwise) to your conjecture that there are no other universal natural besides those stated above. $\endgroup$ – Jack L. Oct 15 at 20:06

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