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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)


A covering system with least modulus 25

Author: Donald Jason Gibson
Journal: Math. Comp. 78 (2009), 1127-1146
MSC (2000): Primary 11B25; Secondary 11A07, 11B75
Published electronically: September 10, 2008
Supplement: Table supplement to this article.
MathSciNet review: 2476575
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Abstract: A collection of congruences with distinct moduli, each greater than $ 1$, such that each integer satisfies at least one of the congruences, is said to be a covering system. A famous conjecture of Erdös from 1950 states that the least modulus of a covering system can be arbitrarily large. This conjecture remains open and, in its full strength, appears at present to be unattackable. Most of the effort in this direction has been aimed at explicitly constructing covering systems with large least modulus. Improving upon previous results of Churchhouse, Krukenberg, Choi, and Morikawa, we construct a covering system with least modulus $ 25$. The construction involves a large-scale computer search, in conjunction with two general results that considerably reduce the complexity of the search.

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Additional Information

Donald Jason Gibson
Affiliation: Eastern Kentucky University, 313 Wallace Building, 521 Lancaster Avenue, Richmond, Kentucky 40475-3102

PII: S 0025-5718(08)02154-6
Received by editor(s): August 27, 2007
Received by editor(s) in revised form: February 23, 2008
Published electronically: September 10, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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