Covering systems with big moduli
Let denote the residue class . A covering system (defined by Paul Erdos in early 1930's) is a finite collection 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 on the size of moduli: e.g. is such system for . 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 , then 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.
* indicates original appearance(s) of problem.