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A new necessary condition for moduli of non-natural irreducible disjoint covering system. (English) Zbl 0822.11004

A disjoint covering system (DCS) is a partition of the set of all integers into a finite number of arithmetic progressions. Non-natural disjoint covering systems are systems of disjoint arithmetic progressions which cannot arise from the set of integers by successive splitting of arithmetic progressions into disjoint ones with greater moduli. Finally, irreducible DCS (IDCS) are those DCS in which no union of its arithmetical progressions is again an arithmetical progression.
The author proves that if \(p^ \alpha\) (\(p\) a prime, \(\alpha>0\)) divides the l.c.m. of moduli in a non-natural IDCS then it divides at least 3 distinct moduli of its arithmetical progressions. Then he shows that if \(m= d_ 1 d_ 2 d_ 3 d_ 4\) is a decomposition of \(m\) into four coprime factors, where only \(d_ 4\) may be equal to 1, then there exists an IDCS with moduli \(d_ 1 d_ 2 d_ 3\), \(d_ 1 d_ 2 d_ 4\), \(d_ 1 d_ 3 d_ 4\), \(d_ 2 d_ 3 d_ 4\). This shows that the above result is the best possible in general. This implies a previous result of I. Korec [ibid. 46/47, 75-81 (1985; Zbl 0612.10002)] and extends a partial answer of H. Kellerer and G. Wirsching [Discrete Math. 85, 191-206 (1990; Zbl 0733.05030)] about ICDS without supremum.

MSC:

11A07 Congruences; primitive roots; residue systems
11B25 Arithmetic progressions
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