Covering systems with big moduli

Importance: Medium ✭✭
Keywords: covering system
Recomm. for undergrads: no
Posted by: Robert Samal
on: August 4th, 2007
Problem   Does for every integer $ N $ exist a covering system with all moduli distinct and at least equal to~$ N $?

Let $ a(n) $ denote the residue class $ \{a+nt \mid t \in \Z\} $. A covering system (defined by Paul Erdos in early 1930's) is a finite collection $ \{a_1(n_1), \dots, a_k(n_k) \} $ of residue classes whose union covers all the integers.

Such systems are easy to find if the moduli are allowed to repeat. They are known for many lower bounds $ N $ on the size of moduli: e.g. $ \{0(2), 0(3), 1(4), 5(6), 7(12) \} $ is such system for $ N=2 $. Choi proved that it is possible to give an example for N = 20.

On the other hand, recently it was shown [FFKPY] that if such systems exist for arbitrary large $ N $, then $ \sum_{i=1}^k \frac 1{n_i} $ is not bounded.


[FFKPY] Michael Filaseta, Kevin Ford, Sergei Konyagin, Carl Pomerance, Gang Yu: Sieving by large integers and covering systems of congruences, J. Amer. Math. Soc. 20 (2007), 495-517.

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