Incongruent restricted disjoint covering systems (Q1043970)

The authors introduce a new type of a covering system, incongruent restricted one (abbreviated IRDCS) on \([1,n]=\{1,2,\dots,n\}\) as a set of at least two congruence classes \(S(m,a)=\{x:x\equiv a\pmod{m}\}\) possessing the property that every integer in the interval \([1,n]\) belongs to exactly one class while each class contains at least two elements of the interval. If \(\{S(m_1,a_1),S(m_2,a_2),\dots,S(m_t,a_t)\}\) then \(n\) is called its length, \(t\) its order and \(\sum_{i=1}^t m_i^{-1}\) its heft. The author report some computational and structural results and present some open problems concerning IRDCS's. For instance, there exist IRDCS of all lengths \(>16\), and one open problem asks whether the smallest modulus of an IRDCS can be arbitrary large.