Sieving by large integers and covering systems of congruences
By Michael Filaseta, Kevin Ford, Sergei V. Konyagin, Carl Pomerance, Gang Yu. 2005
51 citations recorded by the source corpus.
A finite set of congruences, where each integer satisfies at least one them, is called a covering system. A famous problem of Erd?s from 1950 [4] is to determine whether for every N there is a covering system with distinct moduli greater than N. In other words, can the minimum modulus in a covering system with distinct moduli be arbitrarily large? In regards to this problem, Erd?s writes in [6], "This is perhaps my favourite problem." It is easy to see that in a covering system, the reciprocal sum of the moduli is at least 1. Examples with distinct moduli are known with least modulus 2, 3, and 4, where this reciprocal sum can be arbitrarily close to 1; see [10], ?F13. Erd?s and Selfridge [5] conjectured that this fails for all large enough choices of the least modulus. In fact, they made the following much stronger conjecture.
Subfields: Algebra and Number Theory, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/d67598f08259cc28428c3017ef65671457661ae0
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In Computational Theory and Mathematics
Citations in the corpus
- Solution of the minimum modulus problem for covering systems (2013, background): “Thus far our reasoning in the case i = 0 roughly follows the treatment of [11], but now we diverge.”