The general case on the order of appearance of product of consecutive Fibonacci and Lucas numbers

2017/07/29 by Khaochim, Narissara, Pongsriiam, Prapanpong
#11B39 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1707.09512

Abstract

Let F_n and L_n be the nth Fibonacci and Lucas number, respectively. For each positive integer m, 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. Recently, D. Marques has obtained a formula for z(F_nF_n+1), z(F_nF_n+1F_n+2), and z(F_nF_n+1F_n+2F_n+3). In this paper, we extend Marques' result to the case z(F_nF_n+1⋯ F_n+k) for every 4≤ k ≤ 6. We also give a formula for z(L_nL_n+1⋯ L_n+k) when k = 5,6 which extends the recent result of Marques and Trojovský. Our method gives a general idea on how to obtain the formulas for z(F_nF_n+1⋯ F_n+k) and z(L_nL_n+1⋯ L_n+k) for every k≥ 1.

Related

• The order of appearance of the product of the first and second Lucas numbers
• Fibonacci or Lucas numbers that are products of two Lucas numbers or two Fibonacci numbers
• Dynamics of the Fibonacci Order of Appearance Map
• On the Sum of the Sixth Powers of Fibonacci Numbers
• The Order of the Fibonacci and Lucas Numbers