The order of appearance of the product of five consecutive Lucas numbers

Abstract

Let $ F_n$ be the $n$th Fibonacci number and let $L_n$ be the $n$th 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 $F_k$. For instance, $z(F_n)=n=z(L_n)/2$, for all $n>2$. In this paper, among other things, we prove that
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$z(L_{n}L_{n+1}L_{n+2}L_{n+3}L_{n+4})=\dfrac{n(n+1)(n+2)(n+3)(n+4)}{12}$,
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for all positive integers $n\equiv 0,8\pmod{12}$.