A COMPLETE FORMULA FOR THE ORDER OF APPEARANCE OF THE POWERS OF LUCAS NUMBERS

Title & Authors
A COMPLETE FORMULA FOR THE ORDER OF APPEARANCE OF THE POWERS OF LUCAS NUMBERS
Pongsriiam, Prapanpong;

Abstract
Let $\small{F_n}$ and $\small{L_n}$ be the nth Fibonacci number and Lucas number, respectively. The order of appearance of m in the Fibonacci sequence, denoted by z(m), is the smallest positive integer k such that m divides $\small{F_k}$. Marques obtained the formula of $\small{z(L^k_n)}$ in some cases. In this article, we obtain the formula of $\small{z(L^k_n)}$ for all $\small{n,k{\geq}1}$.
Keywords
Fibonacci number;Lucas number;the order of appearance;the rank of appearance;
Language
English
Cited by
References
1.
B. A. Brousseau, Fibonacci and Related Number Theoretic Tables, Santa Clara, The Fibonacci Association, 1972.

2.
J. H. Halton, On the divisibility properties of Fibonacci numbers, Fibonacci Quart. 4 (1966), no. 3, 217-240.

3.
T. Koshy, Fibonacci and Lucas Numbers with Applications, New York, Wiley, 2001.

4.
T. Lengyel, The order of the Fibonacci and Lucas Numbers, Fibonacci Quart. 33 (1995), no. 3, 234-239.

5.
D. Marques, The order of appearance of powers of Fibonacci and Lucas numbers, Fi-bonacci Quart. 50 (2012), no. 3, 239-245.

6.
D. Marques, The order of appearance of integers at most one away from Fibonacci numbers, Fibonacci Quart. 50 (2012), no. 1, 36-43.

7.
P. Pongsriiam, Exact divisibility by powers of the Fibonacci and Lucas numbers, J. Integer Seq. 17 (2014), no. 11, Article 14.11.2, 12 pp.

8.
S. Vajda, Fibonacci and Lucas Numbers and the Golden Section: Theory and Applications, New York, Dover Publications, 2007.