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Let us call a set $D\subseteq\mathbb Z$ residue-class dense if for each residue class $[a]_n=\{kn+a\mid k\in\mathbb Z\}$, there is a residue class $[b]_m$ with $[b]_m\subseteq [a]_n\cap D$.

Using the Sun-tzu (Chinese) Remainder Theorem we can see that examples of dense sets include the non-primes and the non-squares. Moreover, if $D_1$ and $D_2$ are residue-class dense then so is $D_1\cap D_2$.

Elements of such dense sets can be thought of as generic integers. Thus, a generic integer is not prime, and not square.

I'm curious if this is a well-studied notion, under another name?

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    $\begingroup$ You don't say what $m$ is. If you would replace whole residue classes $[b]_m$ in your definition by individual integers $b$ in $[a]_n \cap D$ then $D$ would be a dense subset of $\mathbf Z$ for the profinite topology. $\endgroup$
    – KConrad
    Commented Dec 5, 2021 at 22:47
  • $\begingroup$ @KConrad thanks for the comment. $m$ is such that $[b]_m\subseteq [a]_n$ which implies that $n$ divides $m$, I guess. $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented Dec 5, 2021 at 23:49
  • $\begingroup$ @KConrad perhaps my definition is equivalent to "comeager under the profinite topology". $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented Dec 6, 2021 at 0:12
  • $\begingroup$ Comeager = residual (which would then be a very evocative name here) $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented Dec 6, 2021 at 2:35

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