A non-analytic proof of the newman—znám result for disjoint covering systems

Abstract

A direct combinatorial proof is given to a generalization of the fact that the largest modulusN of a disjoint covering system appears at leastp times in the system, wherep is the smallest prime dividingN. The method is based on geometric properties of lattice parallelotopes.

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References

  1. [1]

    M. A. Berger, A. Felzenbaum andA. S. Fraenkel, New results for covering systems of residue sets,Bull. Amer. Math. Soc. 14 (1986), 121–125.

    MATH  MathSciNet  Article  Google Scholar 

  2. [2]

    P. Erdős, On a problem concerning systems of congruences (Hungarian; English summary),Mat. Lapok. 3 (1952), 122–128.

    MathSciNet  Google Scholar 

  3. [3]

    M. Newman, Roots of unity and covering sets,Math. Ann. 191 (1971), 279–282.

    Article  MathSciNet  Google Scholar 

  4. [4]

    Š. Porubský,Results and Problems on Covering Systems of Residue Classes, Mitteilungen aus dem Math. Sem. Giessen, Heft 150, Universität Giessen, 1981.

  5. [5]

    Š. Znám, On exactly covering systems of arithmetic sequences,Number Theory, Colloq. Math. Societatis János Bolyai2 (P. Turán, ed.), Debrecen 1968, North-Holland, Amsterdam 1970, 221–225.

    Google Scholar 

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This research was supported by grant 85-00368 from the United States-Is rael Binational Science Foundation, Jerusalem, Israel.

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Berger, M.A., Felzenbaum, A. & Fraenkel, A.S. A non-analytic proof of the newman—znám result for disjoint covering systems. Combinatorica 6, 235–243 (1986). https://doi.org/10.1007/BF02579384

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