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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Covering the set of integers by congruence classes of distinct moduli
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by S. L. G. Choi PDF
Math. Comp. 25 (1971), 885-895 Request permission

Abstract:

A set of congruences is called a covering set if every integer belongs to at least one of the congruences. Erdös has raised the following question: given any number N, does there exist a covering set of distinct moduli such that the least of such moduli is N. This has been answered in the affirmative for N up to 9. The aim of this paper is to show that there exists a covering set of distinct moduli the least of which is 20. Recently, Krukenberg independently and by other methods has also obtained results up through $N = 18$.
References
    P. Erdös, Quelques Problèmes de la Théorie des Nombres, Monographies de L’Enseignement Mathématique, no. 6, L’Enseignement Mathématique, Université de Genève, 1963, pp. 81-135. MR 28 #2070.
  • R. F. Churchhouse, Covering sets and systems of congruences, Computers in Mathematical Research, North-Holland, Amsterdam, 1968, pp. 20–36. MR 0240045
  • C. E. Krukenberg, Ph.D. Thesis, University of Illinois, Urbana-Champaign, Ill., 1971, pp. 38-77.
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Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Math. Comp. 25 (1971), 885-895
  • MSC: Primary 10A10
  • DOI: https://doi.org/10.1090/S0025-5718-1971-0297692-7
  • MathSciNet review: 0297692