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I've tried seeking this out in a few places and haven't come up with much that's useful.

(Edit: based on suggestion in the comments, I'm restricting the problem to coprime moduli. A restriction to only prime moduli might help more and would still be useful.)

The standard CRT gives one congruence for each modulus, and gives a unique answer modulo the product of the original moduli (for prime moduli). But what about a setup where you have a set of answers based on sets of congruences/residues? An example, where X is the set of results:

$$ x \in X \iff \begin{cases} x \equiv 0, 1, \text{or } 2 \pmod 5 \\ x \equiv 2, 4, \text{or } 6 \pmod 7 \\ x \equiv 3, 5, \text{or } 8 \pmod{11} \end{cases} $$

In this case, $$X = \{16, 25, 27, 30, 41, 60, 102, 107, 135, 137, 146, 151, 170, 181, 195,\\ 212, 247, 256, 261, 272, 291, 300, 305, 335, 366, 377, 382\}\pmod{385}$$

And of course the results set must have 27 members (3x3x3).

Are there any theorems or results for this sort of problem? Anything like:

  • There must be some subset of solutions between M and N
  • The [smallest/largest] solution must be [less than/greater than] M
  • Certain patterns in the congruences create certain patterns in the results

Or am I barking up an unsolved/uninteresting/proven unsolvable tree?

Bernard
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Eric Snyder
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    @Will I see nothing in the OP indicating that they are considering noncoprime moduli (in fact "unique solutiom" implies coprime). so you comment does not seem to be relevant. Update: OP just edited it to explicitly specify coprime moduli. – Bill Dubuque Sep 23 '20 at 22:01
  • I'm uncertain what you mean by this... yes, a single value for each modulus gives a unique answer, but you're talking about enumerating the entire set of answers, which doesn't hit on what I'm looking at. – Eric Snyder Sep 23 '20 at 22:02
  • @BillDubuque I edited the OP to indicate coprime moduli, a useful constraint and better to specify it than to leave it open. – Eric Snyder Sep 23 '20 at 22:03
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    There's not much that can be said in such generality. Sometimes these problems are solved by sieving, e.g. see the links here. – Bill Dubuque Sep 23 '20 at 22:05
  • @BillDubuque Thanks for the link! Sadly the link to the Wooding thesis you link is expired. I'll see if I can find it elsewhere. Edit: Found it! Not sure how to make a good link though. OK, link here. – Eric Snyder Sep 23 '20 at 22:10
  • @Eric Happily you can search for it and locate it, and please update the link after doing so. – Bill Dubuque Sep 23 '20 at 22:14
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    I think the Lehmer paper may well be my best start. @WillJagy, still entirely uncertain how you feel the Pinasco paper applies to this? – Eric Snyder Sep 23 '20 at 22:24
  • @Eric Yes, if you chase citations to the classic paper of Lehmer that I cited then you will likely locate more recent work iirc. – Bill Dubuque Sep 23 '20 at 22:47

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