Geodesic
The general case on the order of appearance of product of consecutive Lucas numbers
Prapanpong Pongsriiam1 ; Narissara Khaochim2 ; Prapanpong Pongsriiam ; Narissara Khaochim
1Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom, Thailand
2Department of Mathematics, Texas A&M University, College Station, USA
Acta mathematica Universitatis Comenianae, Tome 87 (2018) no. 2, pp. 277-289 
Prapanpong Pongsriiam; Narissara Khaochim; Prapanpong Pongsriiam; Narissara Khaochim. The general case on the order of appearance of product of consecutive Lucas numbers. Acta mathematica Universitatis Comenianae, Tome 87 (2018) no. 2, pp. 277-289. http://geodesic.mathdoc.fr/item/AMUC_2018_87_2_a10/
@article{AMUC_2018_87_2_a10,
     author = {Prapanpong Pongsriiam and Narissara Khaochim and Prapanpong Pongsriiam and Narissara Khaochim},
     title = { The general case on the order of appearance of product of consecutive {Lucas} numbers},
     journal = {Acta mathematica Universitatis Comenianae},
     pages = {277--289},
     year = {2018},
     volume = {87},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/AMUC_2018_87_2_a10/}
}
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Voir la notice de l'article provenant de la source Comenius University

Let $F_n$ and $L_n$ 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 $F_k$. The formula for $z(L_nL_{n+1}L_{n+2}\cdots L_{n+k})$ has been recently obtained by Marques for $1\leq k \leq 3$ and by Marques and Trojovsky for $k = 4$. In this article, we extend the results to the cases $k = 5$ and $k = 6$. Our method gives a general idea on how to obtain the formulas of $z(L_{n}L_{n+1}\cdots L_{n+k})$ for every $k\geq 1$.