Covers of the integers with odd moduli and their applications to the forms 𝑥^{𝑚}-2ⁿ and 𝑥²-𝐹_{3𝑛}/2

Wu, Ke-Jian; Sun, Zhi-Wei

doi:10.1090/S0025-5718-09-02212-1

Covers of the integers with odd moduli and their applications to the forms $x^{m}-2^{n}$ and $x^{2}-F_{3n}/2$
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by Ke-Jian Wu and Zhi-Wei Sun;

Math. Comp. 78 (2009), 1853-1866

DOI: https://doi.org/10.1090/S0025-5718-09-02212-1

Published electronically: January 30, 2009

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Abstract:

In this paper we construct a cover $\{a_{s}(\operatorname {mod} \ n_{s})\}_{s=1}^{k}$ of $\mathbb {Z}$ with odd moduli such that there are distinct primes $p_{1},\ldots ,p_{k}$ dividing $2^{n_{1}}-1,\ldots ,2^{n_{k}}-1$ 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 $a,n\in \{0,1,2,\ldots \}$ and $p$ a prime. We also construct another cover of $\mathbb {Z}$ with odd moduli and use it to prove that $x^{2}-F_{3n}/2$ has at least two distinct prime factors whenever $n\in \{0,1,2,\ldots \}$ and $x\equiv a (\operatorname {mod} M)$, where $\{F_{i}\}_{i\geqslant 0}$ is the Fibonacci sequence, and $a$ and $M$ are suitable positive integers having 80 decimal digits.

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