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Let $S\subsetneq\mathbb{Z}$ be an infinite set. Does there always exist a prime $p$ such that the closure of $S$ in the $p$-adic integers, $\mathbb{Z}_p$, contains a rational integer $n\notin S$?

Or, in elementary language, does there always exist a prime $p$ and $n\in \mathbb{Z}\setminus S$ such that for all $k$ there is an $s\in S$ such that $p^k\mid n-s$.

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    Okay, I $m$oved it to an answer.2011-05-21

1 Answers 1

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How about $S = \{ 0 \} \cup \{ n! : n \in \mathbb{N} \}$?