On infinite disjoint covering systems

Authors:
Aviezri S. Fraenkel and R. Jamie Simpson

Journal:
Proc. Amer. Math. Soc. **119** (1993), 5-9

MSC:
Primary 11B25

MathSciNet review:
1148023

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Abstract: The structure of all infinite incongruent disjoint covering systems (IIDCS) whose moduli are divisible by no prime is given. It is then shown that this structure characterizes the subset of IIDCS for which the greatest common factor of all the moduli is , and the set of primes dividing the moduli is finite.

**[1]**John Beebee,*Examples of infinite, incongruent exact covers*, Amer. Math. Monthly**95**(1988), no. 2, 121–123. MR**935423**, 10.2307/2323066**[2]**M. A. Berger, A. Felzenbaum, A. S. Fraenkel, and R. Holzman,*On infinite and finite covering systems*, Amer. Math. Monthly**98**(1991), no. 8, 739–742. MR**1130685**, 10.2307/2324427**[3]**James Dewar,*On finite and infinite covering sets*, Proceedings of the Washington State University Conference on Number Theory (Washington State Univ., Pullman, Wash., 1971) Dept. Math., Washington State Univ., Pullman, Wash., 1971, pp. 201–206. MR**0318048****[4]**C. E. Krukenberg,*Covering sets of the integers*, Ph.D. Thesis, Univ. of Illinois, Urbana-Champaign, IL, 1971.**[5]**R. J. Simpson and Doron Zeilberger,*Necessary conditions for distinct covering systems with square-free moduli*, Acta Arith.**59**(1991), no. 1, 59–70. MR**1133237****[6]**Sherman K. Stein,*Unions of arithmetic sequences*, Math. Ann.**134**(1958), 289–294. MR**0093493****[7]**Štefan Znám,*A survey of covering systems of congruences*, Acta Math. Univ. Comenian.**40(41)**(1982), 59–79 (English, with Russian and Slovak summaries). MR**686961**

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DOI:
https://doi.org/10.1090/S0002-9939-1993-1148023-8

Article copyright:
© Copyright 1993
American Mathematical Society