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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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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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

MR Author ID:
254588

Email:
zwsun@nju.edu.cn

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.