Infinite covering systems of congruences which don't exist
Author:
Ethan Lewis
Journal:
Proc. Amer. Math. Soc. 124 (1996), 355360
MSC (1991):
Primary 11B25; Secondary 11A07
MathSciNet review:
1301513
Fulltext PDF Free Access
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Abstract: We prove there is no infinite set of congruences with: every integer satisfying exactly one congruence, distinct moduli, the sum of the reciprocals of the moduli equal to 1, the lcm of the moduli divisible by only finitely many primes, and a prime greater than 3 dividing any of the moduli.
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 J. Beebee, Examples of infinite, incongruent exact covers, Amer. Math. Monthly 95 (1988), MR 89g:11013.
 2
 M. A. Berger, A. Felzenbaum, and A. S. Fraenkel, New results for covering systems of residue sets, Bull. Amer. Math. Soc. (N.S.) 14 (1986), 121125, MR 87c:11013.
 3
 M. A. Berger, A. Felzenbaum, A. S. Fraenkel, and R. Holzman, On infinite and finite covering systems, Amer. Math. Monthly 98 (1991), 739742, MR 92g:11009.
 4
 A. S. Fraenkel and R. J. Simpson, On infinite disjoint covering systems, Proc. Amer. Math. Soc. 119 (1993), 59, MR 93k:11006.
 5
 R. K. Guy, Unsolved problems in number theory, Springer, New York, 1981, MR 83k:10002.
 6
 H. Halberstam and K. F. Roth, Sequences, Springer, New York, 1983, MR 83m:10094.
 7
 C. E. Krukenberg, Covering sets of the integers, Univ. of Illinois UrbanaChampaign, 1971.
 8
 W. J. Leveque, Fundamentals of number theory, AddisonWesley, Reading, MA, 1977, MR 58:465.
 9
 S. Porubský, Results and problems on covering systems of residue classes, Mitteilungen aus dem Math. Sem. Giessen, Heft 150, Unitersität Giessen, 1981, MR 83b:10068.
 10
 H. L. Royden, Real analysis, Macmillan, New York, 1988, MR 90g:00004.
 11
 R. J. Simpson, Exact coverings of the integers by arithmetic progressions, Discrete Math. 59 (1986), 181190, MR 87f:11011.
 12
 , Disjoint covering systems of congruences, Amer. Math. Monthly 94 (1987), 865868, MR 89b:11006.
 13
 R. J. Simpson and D. Zeilberger, Necessary conditions for distinct covering systems with squarefree moduli, Acta Arith. 59 (1991), 5970, MR 92i:11014.
 14
 S. K. Stein, Unions of arithmetic sequences, Math Ann. 134 (1958) 282294, MR 20:17.
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 C. Vanden Eynden, On a problem of Stein concerning infinite covers, Amer. Math. Monthly 99 (1992), 355358, MR 93b:11004.
 16
 D. Zeilberger, On a conjecture of R. J. Simpson about exact covering congruences, Amer. Math. Monthly 96 (1989), 243, MR 90a:11008.
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Additional Information
Ethan Lewis
Affiliation:
Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 191046395
Email:
ethan@thales.math.upenn.edu
DOI:
http://dx.doi.org/10.1090/S0002993996030882
PII:
S 00029939(96)030882
Received by editor(s):
December 9, 1992
Received by editor(s) in revised form:
April 18, 1994, and August 20, 1994
Additional Notes:
Supported in part by DOE grant P200A20337.
Communicated by:
William W. Adams
Article copyright:
© Copyright 1996
American Mathematical Society
