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I have a question. Using the Chinese Remainder Theorem, it's easy to find the (lowest) solution to a series of modulus statements. For instance, given:

x mod 5 = 3. x mod 7 = 2. x mod 11 = 9.

the solution (163) is fairly straightforward to find. However, suppose rather than one set of possibilities, we have a group, such as:

x mod 11 = 2,3,4,7,9, or 10. x mod 13 = 0,1,6,7,8, or 9. x mod 17 = 4,5,6,9,10, or 12. x mod 19 = 2,4,6,10,14, or 16.

The set of solutions has 6*6*6*6 = 1,296 members, but I can't find anything about these types of sets. Do they have a name, or is much known about them? (For instance, if I wanted to find the lowest member of the set, or the two closest members, is an efficient way of doing so known?)

user403066
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  • I think modular math gets up to ring theory not just sets if a set is equipped with an operator and it's closed under it it's a magma at very least and then so on in group theory etc. things tend to get called the highest thing they fit in the ladder at last check. –  Aug 09 '17 at 22:19
  • Search on "Lehmer sieve" to find literature on this and related problems. For starters see D. H. Lehmer, The sieve problem for all-purpose computers. – Bill Dubuque Aug 09 '17 at 22:51

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