A polynomial investigation inspired by work of Schinzel and Sierpiński

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] considered the analogous problem of finding f(x) ∈ Z[x] such