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

TL;DR

This paper extends existing formulas for the order of appearance of products of consecutive Fibonacci and Lucas numbers, providing new results for longer products and a general approach for all lengths.

Contribution

It generalizes previous formulas to longer products of Fibonacci and Lucas numbers and introduces a method applicable to all product lengths.

Findings

Formulas for z(F_n F_{n+1} ... F_{n+k}) for 4 ≤ k ≤ 6

Formulas for z(L_n L_{n+1} ... L_{n+k}) for k=5,6

A general method for arbitrary product lengths

Abstract

Let $F_{n}$ and $L_n$ be the $n$th 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 $z(F_{n}F_{n+1})$, $z(F_{n}F_{n+1}F_{n+2})$, and $z(F_{n}F_{n+1}F_{n+2}F_{n+3})$. In this paper, we extend Marques' result to the case $z(F_{n}F_{n+1}\cdots F_{n+k})$ for every $4\leq k \leq 6$. We also give a formula for $z(L_nL_{n+1}\cdots L_{n+k})$ when $k = 5,6$ which extends the recent result of Marques and Trojovsk\'y. Our method gives a general idea on how to obtain the formulas for $z(F_nF_{n+1}\cdots F_{n+k})$ and $z(L_nL_{n+1}\cdots L_{n+k})$ for every $k\geq 1$.