Covers of the integers with odd moduli and their applications to the forms $x^m-2^n$ and $x^2-F_{3n}/2$

Abstract

In this paper we construct a cover {a_s(mod n_s)}_{s=1}^k of Z with odd moduli such that there are distinct primes p_1,...,p_k dividing 2^{n_1}-1,...,2^{n_k}-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 m-th powers of whose terms are never of the form $2^n\pm p^a$ with p a prime and a,n in {0,1,2,...}. We also construct another cover of Z with odd moduli and use it to prove that $x^2-F_{3n}/2$ has at least two distinct prime factors whenever n is a nonnegative integer and x=a (mod M), where {F_i}_{i\ge 0} is the Fibonacci sequence, and a and M are suitable positive integers having 80 decimal digits.