On the order of appearance of the difference of two Lucas numbers

Pavel Trojovsky;


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, $z(L_n)=2n$, for all $n>2$. In this paper, among other things, we prove that \begin{center} $z(L_m-L_n)=\dfrac{5F_p}{p}\cdot \dfrac{m^2-n^2}{4}$, \end{center} for all distinct positive integers $m\equiv n\pmod 4$, with $\gcd(m,n)=p>2$ prime.

Vol. 19 (2018), No. 1, pp. 641-648

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