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Source. Theorem 2, printed p. 4 (PDF p. 5) of the 22 May 2001 author manuscript.

Statement

Let $d$ be an odd positive integer. Suppose there is an $f(x)\in\mathbb Z[x]$ such that

$$f(1)\neq-d$$

and $f(x)x^n+d$ is reducible for every integer $n\geq0$ (over the rationals, as the paper's problem is posed on p. 1). Then an odd covering of the integers exists: a finite covering whose moduli are distinct odd integers greater than $1$.

Proof pointer. The manuscript describes this as a variation of Schinzel's argument and proves it in Section 5, printed pp. 19--22 (PDF pp. 20--23). That proof was not reconstructed or independently checked here.

Bears on. This is a conditional implication toward Problem 7, not an existence theorem for the required polynomial or covering.