Covering systems of homogeneous congruences (Q1293520)

A set of ordered triples \(\{(a_i,b_i,m_i): i=1,\dots,t\}\) with \(a_i, b_i, m_i\) coprime such that every ordered pair of integers \((x,y)\) satisfies \(a_x+b_iy\equiv 0\pmod{m_i}\) for at least one value of \(i\) is called a homogeneous covering system of \(\mathbb{Z}^2\). \textit{T. Cochrane} and \textit{G. Myerson} [Rocky Mt. J. Math. 26, 77-81 (1996; Zbl 0858.11005)] found a method how to construct such systems by using an incongruent covering system of \(\mathbb{Z}\) with each modulus composite. In the present note a method is described how to construct homogeneous covering systems without using such composite systems. The core of the method is based on two processes called here splitting and consolidation (and actually already known for one dimensional covering systems of \(\mathbb{Z}\)). It is also proved here that \((\alpha+1)(p-1)+2\) is the least number of homogeneous congruences \((a_i,b_i,p^{\alpha+1})\) each having modulus \(p^{\alpha+1}\) that cover \((1,b,p^\alpha)=\{(x,y):x+by\equiv 0\pmod{p^\alpha}\}\).