A covering system with least modulus 25

Gibson, Donald

doi:10.1090/S0025-5718-08-02154-6

A covering system with least modulus 25
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by Donald Jason Gibson;

Math. Comp. 78 (2009), 1127-1146

DOI: https://doi.org/10.1090/S0025-5718-08-02154-6

Published electronically: September 10, 2008

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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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