Hostname: page-component-7dd5485656-zklqj Total loading time: 0 Render date: 2025-10-31T04:04:28.033Z Has data issue: false hasContentIssue false
  • English
  • Français

On a conjecture of Crittenden and Vanden Eynden concerning coverings by arithmetic progressions

Part of: Sequences and sets

Published online by Cambridge University Press:  09 April 2009

R. J. Simpson
R. J. Simpson
Affiliation:
School of Mathematics Curtin University of TechnologyPerth, WA 6001Australia e-mail: simpson@cs.curtin.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Crittenden and Vanden Eynden conjectured that if n arithmetic progressions, each having modulus at least k, include all the integers from 1 to k2n-k+1, then they include all the integers. They proved this for the cases k = 1 and k = 2. We give various necessary conditions for a counterexample to the conjecture; in particular we show that if a counterexample exists for some value of k, then one exists for that k and a value of n less than an explicit function of k.

Keywords

arithmetic progressionscovering systems.

MSC classification

Secondary: 11B25: Arithmetic progressions

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 63 , Issue 3 , December 1997 , pp. 396 - 420
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Crittenden, R. B. and Eynden, C. L. Vanden, ‘Any n arithmetic progressions covering the first 2n integers cover all the integers’, Proc. Amer. Math. Soc. 24 (1970), 475–481.Google Scholar
[2]Crittenden, R. B. and Eynden, C. L. Vanden, ‘The union of arithmetic progressions not less than k’, Amer. Math. Monthly 79 (1972), 630.Google Scholar
[3]Erdös, P., ‘Remarks on number theory IV: extremal problems in number theory I’, Mat. Lapok 13 (1962), 241–243 (Hungarian, English summary).Google Scholar
[4]Guy, R. K., Unsolved problems in number theory (Springer, Berlin, 1980), pp. 140–141.Google Scholar
[5]Leveque, W. J., Fundamentals of number theory (Addison-Wesley, Reading, 1977).Google Scholar
[6]Nagura, J., ‘On the interval containing at least one prime number’, Proc. Japan Acad. 28 (1952), 177–181.Google Scholar
[7]Porubsky, S., ‘Results and problems on covering systems of congruences’, Czechoslovak Math. J. 24 (1974), 598–606.CrossRefGoogle Scholar
[8]Rosser, G. Barkley and Schoenfeld, L., ‘Approximate formulas for some functions of prime numbers’, Illinois J. Math. 6 (1962), 64–94.CrossRefGoogle Scholar
[9]Selfridge, J., Research announcement, Amer. Math. Soc. Annual Meeting (New Orleans, 1969).Google Scholar
[10]Simpson, R. J., Covering the integers with arithmetic progressions (Ph. D. Thesis, University of Adelaide, 1982).Google Scholar
[11]Simpson, R. J., ‘Regular coverings the integers by arithmetic progressions’, Acta Arithmetica 45 (1985), 145–152.CrossRefGoogle Scholar
[12]Stein, S. K., ‘Unions of arithmetic sequences’, Math. Ann. 134 (1958), 289–294.CrossRefGoogle Scholar