In this paper we construct a cover {as(mod ns)} k s=1 of Z with odd moduli such that there are distinct primes p1,..., pk dividing 2n1 − 1,..., 2nk − 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 2n ± pa 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 x2 − 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

Year: 2008

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