## 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 PDF
- Math. Comp.
**78**(2009), 1853-1866 Request permission

## 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.## References

- Christian Ballot and Florian Luca,
*On the equation $x^2+dy^2=F_n$*, Acta Arith.**127**(2007), no. 2, 145–155. MR**2289980**, DOI 10.4064/aa127-2-4 - A. S. Bang,
*Taltheoretiske Undersgelser*, Tidsskrift for Mat.**4**(1886), no. 5, 70–80, 130–137. - John Brillhart, D. H. Lehmer, J. L. Selfridge, Bryant Tuckerman, and S. S. Wagstaff Jr.,
*Factorizations of $b^{n}\pm 1$*, Contemporary Mathematics, vol. 22, American Mathematical Society, Providence, R.I., 1983. $b=2,\,3,\,5,\,6,\,7,\,10,\,11,\,12$ up to high powers. MR**715603** - Yann Bugeaud, Maurice Mignotte, and Samir Siksek,
*Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers*, Ann. of Math. (2)**163**(2006), no. 3, 969–1018. MR**2215137**, DOI 10.4007/annals.2006.163.969 - Geo. D. Birkhoff and H. S. Vandiver,
*On the integral divisors of $a^n-b^n$*, Ann. of Math. (2)**5**(1904), no. 4, 173–180. MR**1503541**, DOI 10.2307/2007263 - Yong-Gao Chen,
*On integers of the forms $k^r-2^n$ and $k^r2^n+1$*, J. Number Theory**98**(2003), no. 2, 310–319. MR**1955419**, DOI 10.1016/S0022-314X(02)00051-3 - Fred Cohen and J. L. Selfridge,
*Not every number is the sum or difference of two prime powers*, Math. Comp.**29**(1975), 79–81. MR**376583**, DOI 10.1090/S0025-5718-1975-0376583-0 - John H. E. Cohn,
*Square Fibonacci numbers, etc*, Fibonacci Quart.**2**(1964), 109–113. MR**161819** - Henri Darmon and Andrew Granville,
*On the equations $z^m=F(x,y)$ and $Ax^p+By^q=Cz^r$*, Bull. London Math. Soc.**27**(1995), no. 6, 513–543. MR**1348707**, DOI 10.1112/blms/27.6.513 - P. Erdös,
*On integers of the form $2^k+p$ and some related problems*, Summa Brasil. Math.**2**(1950), 113–123. MR**44558** - Michael Filaseta, Carrie Finch, and Mark Kozek,
*On powers associated with Sierpiński numbers, Riesel numbers and Polignac’s conjecture*, J. Number Theory**128**(2008), no. 7, 1916–1940. MR**2423742**, DOI 10.1016/j.jnt.2008.02.004 - Michael Filaseta, Kevin Ford, Sergei Konyagin, Carl Pomerance, and Gang Yu,
*Sieving by large integers and covering systems of congruences*, J. Amer. Math. Soc.**20**(2007), no. 2, 495–517. MR**2276778**, DOI 10.1090/S0894-0347-06-00549-2 - Song Guo and Zhi-Wei Sun,
*On odd covering systems with distinct moduli*, Adv. in Appl. Math.**35**(2005), no. 2, 182–187. MR**2152886**, DOI 10.1016/j.aam.2005.01.004 - Richard K. Guy,
*Unsolved problems in number theory*, 3rd ed., Problem Books in Mathematics, Springer-Verlag, New York, 2004. MR**2076335**, DOI 10.1007/978-0-387-26677-0 - Hong Hu and Zhi-Wei Sun,
*An extension of Lucas’ theorem*, Proc. Amer. Math. Soc.**129**(2001), no. 12, 3471–3478. MR**1860478**, DOI 10.1090/S0002-9939-01-06234-7 - Kenneth Ireland and Michael Rosen,
*A classical introduction to modern number theory*, 2nd ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. MR**1070716**, DOI 10.1007/978-1-4757-2103-4 - Florian Luca and Pantelimon Stănică,
*Fibonacci numbers that are not sums of two prime powers*, Proc. Amer. Math. Soc.**133**(2005), no. 7, 1887–1890. MR**2099413**, DOI 10.1090/S0002-9939-05-07827-5 - Paulo Ribenboim,
*The little book of bigger primes*, 2nd ed., Springer-Verlag, New York, 2004. MR**2028675** - Zhi Wei Sun,
*Reduction of unknowns in Diophantine representations*, Sci. China Ser. A**35**(1992), no. 3, 257–269. MR**1183711** - Zhi-Wei Sun,
*On integers not of the form $\pm p^a\pm q^b$*, Proc. Amer. Math. Soc.**128**(2000), no. 4, 997–1002. MR**1695111**, DOI 10.1090/S0002-9939-99-05502-1 - Zhi-Wei Sun and Si-Man Yang,
*A note on integers of the form $2^n+cp$*, Proc. Edinb. Math. Soc. (2)**45**(2002), no. 1, 155–160. MR**1884609**, DOI 10.1017/S0013091500000924 - 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** - K. Zsigmondy,
*Zur Theorie der Potenzreste*, Monatsh. Math. Phys.**3**(1892), no. 1, 265–284 (German). MR**1546236**, DOI 10.1007/BF01692444

## 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
- 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.
- © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2501080