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On partitioning \(\mathbb Z_m\) into pairs of prescribed differences. (English) Zbl 1399.05010

Summary: Let \(m=2n\) be a positive integer and \(\mathbb Z_m\) be the ring of residue classes modulo \(m\). Suppose that \(d_1\), \(d_2,\dots, d_n\) are any odd elements (not necessarily distinct) of \(\mathbb Z_m\). In this paper, we give the sufficient and necessary condition of partitioning \(\mathbb Z_m\) into \(n\) pairs with differences \(d_1\), \(d_2,\dots,d_n\). The conjecture on the seating couples problem can be seen as a corollary of our result. Based on this work, we obtain two corollaries and we also confirm a conjecture which is the special case of our first corollary.

MSC:

05A18 Partitions of sets
05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
11B13 Additive bases, including sumsets
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