On homogeneous covering congruences
\textit{T. Cochrane} and \textit{G. Myerson} [Rocky Mt. J. Math. 26, No. 1, 77-81 (1996; Zbl 0858.11005)] call a system of congruences \(\sum_{j=1}^n a_{ij}x_j\equiv 0\pmod{m_i}\), \(1\leq i\leq r\), with \(1<m_1<\dots<m_r\) a homogeneous cover of \(\mathbb{Z}^n\) if every \(n\)--tuple of integers satisfies at least one of the congruences. They proved an intermediate result that given a cover \(a_i\pmod{m_i}\), \(1\leq i\leq r\) of \(\mathbb{Z}\) with all the moduli \(n_i\) composite, the system \((*)\) \(y\equiv 0\pmod{p_j}\), \(x-a_iy\equiv 0\pmod{m_i}\), \(1\leq i\leq r, 1\leq j\leq t\) form a homogeneous cover of \(\mathbb{Z}^2\) with distinct moduli, where \(p_1,\dots,p_t\) are all the primes dividing \(\prod_{i=1}^r m_i\). They posed several questions in that paper: one of them asks whether there is a homogeneous cover of \(\mathbb{Z}^2\) that does not come from a cover of \(\mathbb{Z}\) with distinct composite moduli. The author makes the question precise, introducing the notions of simple transformation and trivial extension. In the first theorem he then proves that there exists a homogeneous cover of \(\mathbb{Z}^2\) with distinct moduli greater than \(1\) that does not come from any cover \((*)\) by a simple transformation followed by a trivial extension. The second main result of the paper asserts that the so-called Erdős conjecture implies that there are homogeneous covers of \(\mathbb{Z}^n\) with distinct moduli greater than \(1\) which do not come from covers of \(\mathbb{Z}^2\) by simple transformations followed be a trivial extension.
scientific article; zbMATH DE number 1051539