Covers of the Integers with Odd Moduli

In this paper we construct a cover {as(mod ns)}s=1 of Z with odd moduli such that there are distinct primes p1, . . . , pk dividing 2 n1 − 1, . . . , 2k − 1 respectively. Using this cover we show that for any positive integer m divisible by none of 3, 5, 7, 11, 13 there exists an infinite arithmetic progression of positive odd integers the mth powers of whose terms are never of the form 2 ± p with a, n ∈ {0, 1, 2, . . . } and p a prime. We also construct another cover of Z with odd moduli and use it to prove that x − F3n/2 has at least two distinct prime factors whenever n ∈ {0, 1, 2, . . . } and x ≡ a (mod M), where {Fi}i>0 is the Fibonacci sequence, and a and M are suitable positive integers having 80 decimal digits.