The notion of covering system was introduced by Paul Erdős in the early 1930s.
A covering system is called distinct (or incongruent) if all the moduli
Hough and Nielsen (2019)[1] proved that any distinct covering system has a modulus that is divisible by either 2 or 3.
A system (i.e., an unordered multi-set) of finitely many residue classes is called an
Another exact cover in common use is that of odd and even numbers, or This is just one case of the following fact: For every positive integer modulus
This result was conjectured in 1950 by Paul Erdős and proved soon thereafter by Leon Mirsky and Donald J. Newman.
The same proof was also found independently by Harold Davenport and Richard Rado.
Erdős's question was resolved in the negative by Bob Hough.
[6] Hough used the Lovász local lemma to show that there is some maximum N<1016 which can be the minimum modulus on a covering system.
There is a famous unsolved conjecture from Erdős and Selfridge: an incongruent covering system (with the minimum modulus greater than 1) whose moduli are odd, does not exist.
It is known that if such a system exists with square-free moduli, the overall modulus must have at least 22 prime factors.