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

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**John Beebee,*Examples of infinite, incongruent exact covers*, Amer. Math. Monthly**95**(1988), no. 2, 121–123. MR**935423**, 10.2307/2323066**2**Marc A. Berger, Alexander Felzenbaum, and Aviezri S. Fraenkel,*New results for covering systems of residue sets*, Bull. Amer. Math. Soc. (N.S.)**14**(1986), no. 1, 121–126. MR**818066**, 10.1090/S0273-0979-1986-15414-5**3**M. A. Berger, A. Felzenbaum, A. S. Fraenkel, and R. Holzman,*On infinite and finite covering systems*, Amer. Math. Monthly**98**(1991), no. 8, 739–742. MR**1130685**, 10.2307/2324427**4**Aviezri S. Fraenkel and R. Jamie Simpson,*On infinite disjoint covering systems*, Proc. Amer. Math. Soc.**119**(1993), no. 1, 5–9. MR**1148023**, 10.1090/S0002-9939-1993-1148023-8**5**Richard K. Guy,*Unsolved problems in number theory*, Unsolved Problems in Intuitive Mathematics, vol. 1, Springer-Verlag, New York-Berlin, 1981. Problem Books in Mathematics. MR**656313****6**Heini Halberstam and Klaus Friedrich Roth,*Sequences*, 2nd ed., Springer-Verlag, New York-Berlin, 1983. MR**687978****7**C. E. Krukenberg,*Covering sets of the integers*, Univ. of Illinois Urbana-Champaign, 1971.**8**William J. LeVeque,*Fundamentals of number theory*, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1977. MR**0480290****9**Štefan Porubský,*Results and problems on covering systems of residue classes*, Mitt. Math. Sem. Giessen**150**(1981), 85. MR**638657****10**H. L. Royden,*Real analysis*, 3rd ed., Macmillan Publishing Company, New York, 1988. MR**1013117****11**R. J. Simpson,*Exact coverings of the integers by arithmetic progressions*, Discrete Math.**59**(1986), no. 1-2, 181–190. MR**837965**, 10.1016/0012-365X(86)90079-8**12**R. J. Simpson,*Disjoint covering systems of congruences*, Amer. Math. Monthly**94**(1987), no. 9, 865–868. MR**935844**, 10.2307/2322820**13**R. J. Simpson and Doron Zeilberger,*Necessary conditions for distinct covering systems with square-free moduli*, Acta Arith.**59**(1991), no. 1, 59–70. MR**1133237****14**Sherman K. Stein,*Unions of arithmetic sequences*, Math. Ann.**134**(1958), 289–294. MR**0093493****15**Charles Vanden Eynden,*On a problem of Stein concerning infinite covers*, Amer. Math. Monthly**99**(1992), no. 4, 355–358. MR**1157227**, 10.2307/2324903**16**Doron Zeilberger,*On a conjecture of R. J. Simpson about exact covering congruences: “Disjoint covering systems of congruences” [Amer. Math. Monthly 94 (1987), no. 9, 865–868; MR0935844 (89b:11006)]*, Amer. Math. Monthly**96**(1989), no. 3, 243. MR**991873**, 10.2307/2325213

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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