Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

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\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.


References [Enhancements On Off] (What's this?)

  • [BL] C. Ballot and F. Luca, On the equation $ x^{2}+dy^{2}=F_{n}$, Acta Arith. 127 (2007), 145-155. MR 2289980 (2008a:11116)
  • [B] A. S. Bang, Taltheoretiske Undersgelser, Tidsskrift for Mat. 4 (1886), no. 5, 70-80, 130-137.
  • [BLSTW] J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagstaff, Jr., Factorizations of $ b^{n}\pm 1$, $ b=2,3,5,6,7,10,11,12$ up to High Powers, Third ed., Contemporary Mathematics 22, Amer. Math, Soc., Providence, RI, 2002. MR 715603 (84k:10005)
  • [BMS] Y. Bugeaud, M. Mignotte and S. Siksek, Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers, Ann. of Math. (2) 163 (2006), 969-1018. MR 2215137 (2007f:11031)
  • [BV] G. D. Birkhoff and H. S. Vandiver, On the integral divisors of $ a^{n}-b^{n}$, Ann. of Math. 5 (1904), 173-180. MR 1503541
  • [C] Y.-G. Chen, On integers of the forms $ k^{r}-2^{n}$ and $ k^{r}2^{n}+1$, J. Number Theory 98 (2003), 310-319. MR 1955419 (2003m:11004)
  • [CS] F. Cohen and J. L. Selfridge, Not every number is the sum or difference of two prime powers, Math. Comput. 29 (1975), 79-81. MR 0376583 (51:12758)
  • [Co] J. H. E. Cohn, Square Fibonacci numbers, etc., Fibonacci Quart. 2 (1964), 109-113. MR 0161819 (28:5023)
  • [DG] H. Darmon and A. Granville, On the equations $ z^{m}=F(x,y)$ and $ Ax^{p}+By^{q}=Cz^{r}$, Bull. London Math. Soc. 27 (1995), 513-544. MR 1348707 (96e:11042)
  • [E] P. Erdős, On integers of the form $ 2^{k}+p$ and some related problems, Summa Brasil. Math. 2 (1950), 113-123. MR 0044558 (13:437i)
  • [FFK] M. Filaseta, C. Finch and M. Kozek, On powers associated with Sierpiński numbers, Riesel numbers and Polignac's conjecture, J. Number Theory 128 (2008), 1916-1940. MR 2423742
  • [FFKPY] M. Filaseta, K. Ford, S. Konyagin, C. Pomerance and G. Yu, Sieving by large integers and covering systems of congruences, J. Amer. Math. Soc. 20 (2007), 495-517. MR 2276778 (2007k:11016)
  • [GS] S. Guo and Z. W. Sun, On odd covering systems with distinct moduli, Adv. in Appl. Math. 35 (2005), 182-187. MR 2152886 (2006e:11018)
  • [Gu] R. K. Guy, Unsolved Problems in Number Theory, Third edition, Springer, New York, 2004, Section A19, B21, F13. MR 2076335 (2005h:11003)
  • [HS] H. Hu and Z. W. Sun, An extension of Lucas' theorem, Proc. Amer. Math. Soc. 129 (2001), 3471-3478. MR 1860478 (2002i:11019)
  • [IR] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Second edition, Grad. Texts in Math. 84, Springer, New York, 1990. MR 1070716 (92e:11001)
  • [LS] F. Luca and P. Stănică, Fibonacci numbers that are not sums of two prime powers, Proc. Amer. Math. Soc. 133 (2005), 1887-1890. MR 2099413 (2005k:11023)
  • [R] P. Ribenboim, The Little Book of Bigger Primes, Second ed., Springer, New York, 2004. MR 2028675 (2004i:11003)
  • [S92] Z. W. Sun, Reduction of unknowns in Diophantine representations, Sci. China Ser. A 35 (1992), no. 3, 257-269. MR 1183711 (93h:11039)
  • [S00] Z. W. Sun, On integers not of the form $ \pm p^{a}\pm q^{b}$, Proc. Amer. Math. Soc. 128 (2000), 997-1002. MR 1695111 (2000i:11157)
  • [SY] Z. W. Sun and S. M. Yang, A note on integers of the form $ 2^{n}+cp$, Proc. Edinburgh Math. Soc. 45 (2002), 155-160. MR 1884609 (2002j:11117)
  • [W] B. M. M. de Weger, Algorithms for Diophantine Equations, CWI Tract, Vol. 65, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1989. MR 1026936 (90m:11205)
  • [Z] K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math. 3 (1892), 265-284. MR 1546236

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11B25, 11A07, 11A41, 11B39, 11D61, 11Y99

Retrieve articles in all journals with MSC (2000): 11B25, 11A07, 11A41, 11B39, 11D61, 11Y99


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.

American Mathematical Society