Covers with Less than 10 Moduli and Their Applications *

In this paper we essentially determine all covers {a s (mod n s)} k s=1 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.