It is possible that a more technical version of this question has been
asked and answered in the literature. If so, then a reference is much
appreciated. I will phrase it in terms of colored tapes placed on a number
line.


I have an unlimited supply of transparent but partially colored 
arithmetic progression tapes,
one for each prime number. Thus, the tape for 3 looks a lot like the tape
for 29, except instead of having a blue square at every third inch, I have a 
yellow square at every 29th inch; the tape is clear otherwise. I am going to 
make several arrangements of tape. My first arrangement produces a color coding
of integers similar to indicating divisibility, except I perversely decide to
line up all the colors at 1. This leaves positions 0 and 2 with no colors, and
every other position gets finitely many colors (88 gets only blue and yellow).


What other arrangements can I make? I can pick any nonempty set of integers
and arrange that each get some of a partition of colors, although the integer may get more colors than I assigned. I can sometimes arrange to leave a square uncolored, regardless of how the primes are partitioned. That is not the
main question though.


Main question: can I arrange that every integer gets only finitely many colors?


An amusing fact (courtesy of the Chinese Remainder Theorem) is that, with a
finite number of such tapes and modulo translation, every such arrangement looks
the same. Thus I do not expect a finite upper bound to the number of colors
received by an integer.


Although I would appreciate a specific example, I am also interested in
knowing how weak a subtheory of ZF or PA is needed to prove existence of
such an arrangement. (I harbor a suspicion that PRA will resolve it
negatively if it resolves at all.) Specific references to the literature
on covering congruences are welcome (as are alternate search terms), but for this problem I am considering
prime moduli only.


Gerhard "Your Responses Are Being Recorded" Paseman, 2014.03.14



Infinite play with tape, or covering the integers with prime arithmetic progressions
There is a map $\varphi: \mathbb{Z}\rightarrow \prod \mathbb{F}\_p$, given by reduction mod $p$ in each coordinate.
If you place the tapes centered at $0$ (to account for divisibility), you can read off the number of colors $n\in \mathbb{Z}$ gets by counting the zero-entries of $\varphi(n)$.


Now in your situation, you shift every tape by a certain amount. This accounts for the choice of an element in $\mathbb{F}\_p$ for each $p$, or an element $\sigma\in \prod \mathbb{F}\_p$.


In your setting, the number of colors associated to $n\in\mathbb{Z}$ will be given by the number of coordinates in which $\varphi(n)$ agrees with $\sigma$.


So we are left with the question of constructing $\sigma\in \prod \mathbb{F}\_p$, which agrees with each of the $\varphi(n)$ only on finitely many digits.


However, this is easy: We can even do an explicit example. Just choose the $p$-th coordinate of $\sigma$ (for example) to be $\lfloor{p/2}\rfloor$. This way $\phi(n)$ will differ from $\sigma$ in the $p$-th coordinate for all $p$ with $p > 2|n|+1$.


The idea is that you place each tape such that $0$ is approximately in the middle of two successive colored spots, this way you get similar behaviour to the $\mathbb{N}$-situation.


Just for the sake of completeness, this explicit example obviously does not rely on any abstract existence arguments, so I guess this proof holds in any reasonable theory.


