of finitely many residue classes whose union covers all the integers.
The notion of covering system was introduced by Paul Erdős in the early 1930s.
A covering system is called disjoint (or exact) if no two members overlap.
A covering system is called distinct (or incongruent) if all the moduli are different (and bigger than 1).
A covering system is called irredundant (or minimal) if all the residue classes are required to cover the integers.
The first two examples are disjoint.
The third example is distinct.
A system (i.e., an unordered multi-set)
of finitely many residue classes is called an -cover if it covers every integer at least times, and an exact -cover if it covers each integer exactly times. It is known that for each there are exactly -covers which cannot be written as a union of two covers. For example,
is an exact 2-cover which is not a union of two covers.
The following problem from Paul Erdős is unsolved: Whether for any arbitrarily large N there exists an incongruent covering system the minimum of whose moduli is at least N. It is easy to construct examples where the minimum of the moduli in such a system is 2, or 3 (Erdős gave an example where the moduli are in the set of the divisors of 120 ); D. Swift gave an example where the minimum of the moduli is 4 (and the moduli are in the set of the divisors of 2880). S. L. G. Choi proved that it is possible to give an example for N = 20.
In another problem we want that all of the moduli (of an incongruent covering system) be odd. 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, the overall modulus must have at least 22 prime factors.