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A polynomial investigation inspired by work of Schinzel and Sierpiński. (English) Zbl 1293.11048
Modifying an idea of A. Schinzel [Acta Arith. 13, 91–101 (1967; Zbl 0171.00701)] and [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 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.

##### MSC:
 11C08 Polynomials in number theory 11A07 Congruences; primitive roots; residue systems 11B25 Arithmetic progressions 11R09 Polynomials (irreducibility, etc.)
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