The Order of Appearance of the Product of Five Consecutive Lucas Numbers

Diego Marques; Pavel Trojovský

doi:10.2478/tmmp-2014-0019

.blurhash-client-img { display: none !important; }

The Order of Appearance of the Product of Five Consecutive Lucas Numbers

Tatra Mountains Mathematical Publications

Volume 59 (2014): Issue 1 (June 2014)

By: Diego Marques and Pavel Trojovský

Open Access

|Mar 2015

Abstract

Let F_n be the nth Fibonacci number and let L_n be the nth Lucas number. The order of appearance z(n) of a natural number n is defined as the smallest natural number k such that n divides Fk. For instance, z(F_n) = n = z(L_n)/2 for all n > 2. In this paper, among other things, we prove that

for all positive integers n ≡ 0,8 (mod 12).

References

[1] BENJAMIN, A.-QUINN, J.: The Fibonacci numbers - Exposed more discretely, Math. Mag. 76 (2003), 182-192.
Search in Google Scholar
[2] HALTON, J. H.: On the divisibility properties of Fibonacci numbers, Fibonacci Quart. 4 (1966), 217-240.
Search in Google Scholar
[3] KALMAN, D.-MENA, R.: The Fibonacci numbers - exposed, Math. Mag. 76 (2003), 167-181.
Search in Google Scholar
[4] KOSHY, T.: Fibonacci and Lucas Numbers with Applications. Wiley, New York, 2001.10.1002/9781118033067
Search in Google Scholar
[5] MARQUES, D.: On integer numbers with locally smallest order of appearance in the Fibonacci sequence, Internat. J. Math. Math. Sci., Article ID 407643 (2011), 4 pages.10.1155/2011/407643
Search in Google Scholar
[6] MARQUES, D.: On the order of appearance of integers at most one away from Fibonacci numbers, Fibonacci Quart. 50 (2012), 36-43.
Search in Google Scholar
[7] MARQUES, D.: The order of appearance of product of consecutive Fibonacci numbers, Fibonacci Quart. 50 (2012), 132-139.
Search in Google Scholar
[8] MARQUES, D.: The order of appearance of powers Fibonacci and Lucas numbers, Fibonacci Quart. 50 (2012), 239-245.
Search in Google Scholar
[9] MARQUES, D.: The order of appearance of product of consecutive Lucas numbers, Fibonacci Quart. 50 (2012), 239-245.
Search in Google Scholar
[10] MARQUES, D.: Sharper upper bounds for the order of appearance in the Fibonacci sequence, Fibonacci Quart. 51 (2013), 233-238.
Search in Google Scholar
[11] LAGARIAS, J. C.: The set of primes dividing the Lucas numbers has density 2/3, Pacific J. Math. 118 (1985), 449-461.10.2140/pjm.1985.118.449
Search in Google Scholar
[12] LENGYEL, T.: The order of the Fibonacci and Lucas numbers, Fibonacci Quart. 33 (1995), 234-239.
Search in Google Scholar
[13] RIBENBOIM, P.: My Numbers, My Friends: Popular Lectures on Number Theory. Springer-Verlag, New York, 2000.
Search in Google Scholar
[14] ROBINSON, D.W.: The Fibonacci matrix modulo m, Fibonacci Quart. 1 (1963), 29-36.
Search in Google Scholar
[15] SALLÉ, H. J. A.: Maximum value for the rank of apparition of integers in recursive sequences, Fibonacci Quart. 13 (1975), 159-161.
Search in Google Scholar
[16] VINSON, J.: The relation of the period modulo m to the rank of apparition of m in the Fibonacci sequence, Fibonacci Quart. 1 (1963), 37-45.
Search in Google Scholar
[17] VOROBIEV, N. N.: Fibonacci Numbers. Birkhäuser, Basel, 2003. 10.1007/978-3-0348-8107-4
Search in Google Scholar

Articles in this issue

DOI: https://doi.org/10.2478/tmmp-2014-0019 | Journal eISSN: 1338-9750 | Journal ISSN: 1210-3195

Journal RSS Feed

Language: English

Page range: 65 - 77

Submitted on: Sep 2, 2014

Published on: Mar 11, 2015

Published by: Slovak Academy of Sciences, Mathematical Institute

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

Related subjects:

© 2015 Diego Marques, Pavel Trojovský, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 59 (2014): Issue 1 (June 2014)