Covers of the integers with odd moduli and their applications to the forms $x^m-2^n$ and $x^2-F_{3n}/2$
| dc.creator | Wu, Ke-Jian | |
| dc.creator | Sun, Zhi-Wei | |
| dc.date | 2007-02-13 | |
| dc.date | 2008-11-28 | |
| dc.date.accessioned | 2026-07-07T12:05:05Z | |
| dc.date.available | 2026-07-07T12:05:05Z | |
| dc.description | In this paper we construct a cover {a_s(mod n_s)}_{s=1}^k of Z with odd moduli such that there are distinct primes p_1,...,p_k dividing 2^{n_1}-1,...,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 p a prime and a,n in {0,1,2,...}. We also construct another cover of Z with odd moduli and use it to prove that $x^2-F_{3n}/2$ has at least two distinct prime factors whenever n is a nonnegative integer and x=a (mod M), where {F_i}_{i\ge 0} is the Fibonacci sequence, and a and M are suitable positive integers having 80 decimal digits. | |
| dc.description | 17 pages; accepted by Mathematics of Computation | |
| dc.identifier | https://arxiv.org/abs/math/0702382 | |
| dc.identifier | http://arxiv.org/abs/math/0702382 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/208285 | |
| dc.subject | Number Theory | |
| dc.subject | Combinatorics | |
| dc.subject | 11B25; 11A07; 11A41; 11B39; 11D61; 11Y99 | |
| dc.title | Covers of the integers with odd moduli and their applications to the forms $x^m-2^n$ and $x^2-F_{3n}/2$ | |
| dc.type | text |