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One of my students asked me the following question. It seemed easy to answer but in fact, I am stucK.

The question: does there exist an infinite subset $S$ of $\mathbb N$ such there exists a positive integer $n_0$ such for all prime $p\ge n_0$ one has $$\mathrm{Card}(S \pmod p)<p$$ where $S\pmod p$ is the subset of classes of $\mathbb Z/p\mathbb Z$ having a representative in $S$.

Thanks in advance (for me and my student :D )

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    $\begingroup$ $S := \{(2n)! : n \in \mathbb{N}\}$ $\endgroup$
    – mathworker21
    Commented Apr 12 at 2:14
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    $\begingroup$ Not a direct answer to this problem, but the square free integers $S$ have the property that $$ \mathrm{Card}(S \pmod p^2)=p^2-1$$ for all primes $p$. Moreover, if I am not mistaken, a set $X$ belongs to the dynamical system generated by $S$ exactly when $$\mathrm{Card}(S \pmod p^2)<p^2$$ for all primes, and I find this last bit very interesting. $\endgroup$
    – Nick S
    Commented Apr 12 at 2:29
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    $\begingroup$ The set of squares also works $\endgroup$
    – Daniel Weber
    Commented Apr 12 at 2:53

1 Answer 1

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If $S=\{p_1,p_1p_2,p_1p_2p_3,\cdots\}$ where $p_n$ is the $n^\text{th}$ prime, then $\operatorname{card}(S\operatorname{mod}p_n)\le n\lt p_n$ for all $n$.

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