A polynomial investigation inspired by work of Schinzel and Sierpiński (Q2919676)

Modifying an idea of \textit{A. Schinzel} [Acta Arith. 13, 91--101 (1967; Zbl 0171.00701)] and [\textit{M. Filaseta}, Number theory for the millennium II. Proceedings of the millennial conference on number theory, Urbana-Champaign, IL, USA, May 21--26, 2000, 1-24 (2002; Zbl 1029.11007)] proved the following result: Let \(d\) be an odd integer. If there is an \(f(x)\in {\mathbb Z}[x]\) satisfying \(f(1)\neq -d\) and \(f(x)\cdot x^n+d\) is reducible over rationals for all \(n\geq 0\), then there exists a finite collection of congruences \(x\equiv a_j\pmod{m_j}\), \(2\not| m_j>1\), with \(1\leq j\leq r\), such that every integer satisfies at least one of these congruences. The existence of such a system of congruences (called odd covering) is a long standing open problem going back to Erdös and Selfridge (cf. (1.9) of the reviewer [Mitt. Math. Semin. Gießen 150, 85 p. (1981; Zbl 0479.10032)]. Improving previous results of the first author [Zbl 1029.11007] and of \textit{L. Jones} [Int. J. Number Theory 5, 999--1015 (2009; Zbl 1231.12001)] the authors prove a related results that if \(d\) is an even integer, then there is an \(f(x)\in {\mathbb Z}[x]\) such that both \(f(1)\neq -d\) and \(f(x)\cdot x^n+d\) is reducible over rational for all \(n\geq 0\). To prove the result it is sufficient to establish it for \(d=2\). It is interesting to note that the authors use 2773 auxiliary congruences to construct 5539 congruences employed in a construction of such a polynomial.