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


Authors: Ke-Jian Wu and Zhi-Wei Sun
Journal: Math. Comp. 78 (2009), 1853-1866
MSC (2000): Primary 11B25; Secondary 11A07, 11A41, 11B39, 11D61, 11Y99
DOI: https://doi.org/10.1090/S0025-5718-09-02212-1
Published electronically: January 30, 2009
MathSciNet review: 2501080
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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\eq 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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Additional Information

Ke-Jian Wu
Affiliation: Department of Mathematics, Zhanjiang Normal University, Zhanjiang 524048, People’s Republic of China
Email: kjwu328@yahoo.com.cn

Zhi-Wei Sun
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China and State Key Laboratory of Novel Software Technology, Nanjing University, Nanjing 210093, People’s Republic of China
Email: zwsun@nju.edu.cn

DOI: https://doi.org/10.1090/S0025-5718-09-02212-1
Keywords: Cover of the integers, arithmetic progression, Fibonacci sequence, prime divisor.
Received by editor(s): February 15, 2007
Received by editor(s) in revised form: July 4, 2008
Published electronically: January 30, 2009
Additional Notes: The second author is responsible for communications, and supported by the National Natural Science Foundation (grant 10871087) of People’s Republic of China.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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