A SHARP RESULT ON m - COVERS 3

Example 1.1. For each integer m > 1, there is an m-cover of Z which is not the union of two covers of Z. To wit, we let p1, . . . , pr be distinct primes with r > 2m − 1, and set N = p1 · · · pr. Clearly A∗ = {∏s∈I ps(N)}I⊆{1,... ,r}, |I|>m does not cover any integer relatively prime to N . Let a1, . . . , an be the list of those integers in {0, 1, . . . , N − 1} not covered by A∗ with each occurring exactly m times. If x ∈ Z is covered by A∗, then x ∈ ⋂ s∈I 0(ps) for some I ⊆ {1, . . . , r} with |I| > m. Therefore