On an Irreducibility Theorem of A. Schinzel Associated with Coverings of the Integers

A covering of the integers is a system of congruences x aj (mod mj), where aj and mj denote integers with mj > 0 for each j, such that every integer satis es at least one of the congruences. An open problem (which surfaced over 40 years ago) is to determine whether a covering of the integers exists for which the indices j range over a nite set and the mj are distinct odd integers > 1. The problem of whether an \odd covering" of the integers, as we will call it, exists led Erd}os and Selfridge to o er money to entice its solution while essentially betting on the outcome of the answer. Erd}os, convinced that an odd covering does exist, o ered $25 for a proof that no odd covering exists; Selfridge, convinced (at that point) that no odd covering exists, o ered $300 for the rst explicit example of an odd covering. No award was promised to someone who gave a non-constructive proof that an odd covering of the integers exists. Over the years, the prize money has varied (cf. [1, p. 251]). Selfridge (private communication) has informed us that he is now increasing his award to $2000. This paper was motivated largely by related work of Schinzel [3] associated with irreducible polynomials. Throughout this paper, unless speci ed otherwise, reducibility and irreducibility shall be in the ring Z[x] (in particular, 1 and 1 are neither reducible nor irreducible). It is well known (based on an appropriate covering argument) that there are in nitely many (even a positive proportion) of positive integers k such that k 2 + 1 is composite for all positive integers n (cf. [1, p. 77]). An analogous problem is to determine whether there exists an f(x) 2 Z[x] such that f(x)x+1 is reducible for all positive integers n. To make the problem non-trivial, one should add the condition that f(1) 6= 1. A consequence of Schinzel's result in [3] is that if there is a polynomial f(x) 2 Z[x] for which f(1) 6= 1 and for which f(x)x + 1 is reducible over the integers for every positive integer n, then there must exist an odd covering of the integers. In fact, Schinzel established that the existence of such an f(x) is equivalent to an explicitly described covering which is more restrictive than an odd covering. The argument he gives is based largely on obtaining speci c knowledge about the factorization of f(x)x + 1 when n is suÆciently large. For the connection between the factorization of f(x)x + 1