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 $N = 18$.
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.
- © 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