On Coverings of the Integers Associated with an Irreducibility Theorem of A. Schinzel
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منابع مشابه
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 probl...
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Define a covering system (or covering) of the integers as a finite collection of congruences x ≡ aj (mod mj), with 1 ≤ j ≤ r, such that every integer satisfies at least one of these congruences. As an interesting application of coverings, W. Sierpiński [4] showed that there are odd positive integers k for which k · 2 + 1 is composite for all integers n ≥ 0. For d ∈ Z, the first author [1] consi...
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is irreducible. Irreducibility here and throughout this paper refers to irreducibility over the rationals. Some condition, such as ja0j = janj = 1, on the integers aj is necessary; otherwise, the irreducibility of all polynomials of the form above would imply every polynomial inZ[x] is irreducible (which is clearly not the case). In this paper, we will mainly be interested in relaxing the condi...
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