Infinite covering systems of congruences

which don't exist

Author:
Ethan Lewis

Journal:
Proc. Amer. Math. Soc. **124** (1996), 355-360

MSC (1991):
Primary 11B25; Secondary 11A07

DOI:
https://doi.org/10.1090/S0002-9939-96-03088-2

MathSciNet review:
1301513

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Abstract | References | Similar Articles | Additional Information

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.

**1**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), 121--125, 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), 739--742, MR**92g:11009**.**4**A. S. Fraenkel and R. J. Simpson,*On infinite disjoint covering systems*, Proc. Amer. Math. Soc.**119**(1993), 5--9, 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 Urbana-Champaign, 1971.**8**W. J. Leveque,*Fundamentals of number theory*, Addison-Wesley, 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), 181--190, MR**87f:11011**.**12**------,*Disjoint covering systems of congruences*, Amer. Math. Monthly**94**(1987), 865--868, MR**89b:11006**.**13**R. J. Simpson and D. Zeilberger,*Necessary conditions for distinct covering systems with square-free moduli*, Acta Arith.**59**(1991), 59--70, MR**92i:11014**.**14**S. K. Stein,*Unions of arithmetic sequences*, Math Ann.**134**(1958) 282--294, MR**20:17**.**15**C. Vanden Eynden,*On a problem of Stein concerning infinite covers*, Amer. Math. Monthly**99**(1992), 355--358, 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 19104-6395

Email:
ethan@thales.math.upenn.edu

DOI:
https://doi.org/10.1090/S0002-9939-96-03088-2

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