Covering the set of integers by congruence classes of distinct moduli

Author:
S. L. G. Choi

Journal:
Math. Comp. **25** (1971), 885-895

MSC:
Primary 10A10

MathSciNet review:
0297692

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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 .

**[1]**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.**[2]**R. F. Churchhouse,*Covering sets and systems of congruences*, Computers in Mathematical Research, North-Holland, Amsterdam, 1968, pp. 20–36. MR**0240045****[3]**C. E. Krukenberg, Ph.D. Thesis, University of Illinois, Urbana-Champaign, Ill., 1971, pp. 38-77.

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DOI:
http://dx.doi.org/10.1090/S0025-5718-1971-0297692-7

Keywords:
Integers,
congruences,
covering set,
distinct moduli

Article copyright:
© Copyright 1971
American Mathematical Society