Covers with Less than 10 Moduli and Their Applications

摘要

In this paper we essentially determine all covers { a_s(mod n_s) }_(s = 1)~k of Z with k < 10, actually our algorithm is valid for any positive integer k. As an application we provide a somewhat general theorem on (infinite) arithmetic progressions (e.g. 1330319+ 346729110 Z) consisting of odd integers no term of which can be expressed as the sum of a power of two and an odd prime, on the other hand we obtain an interesting result on integers of the form 2~n + cp where c is a constant and p is a prime.