Residues of Terms of Lucas Sequences Modulo $3^k$

Abstract

The Fibonacci sequence defined by $F_0=0$, $F_1=1$, and $F_n=F_{n-1}+F_{n-2}$ has a shortest period length of $4\cdot 3^{k-1}$ modulo $3^k$ for every $k\in\mathbb{N}$. In 2011, Bundschuh and Bundschuh \cite{bundschuh3} gave the frequencies of every residue $0\leq b\leq 3^k-1$ in this shortest period. In particular, their result implies that the Fibonacci sequences is not stable modulo $3$. Here we extend this result to other Lucas sequences. More specifically, we give analogous results for Lucas sequences defined by $\left(u_n\right)_n$ with $u_0=0$, $u_1=1$, and $u_n=Pu_{n-1}+u_{n-2}$ for all $n\geq 2$, as well as Lucas sequences defined by $\left(v_n\right)_n$ with $v_0=2$, $v_1=P$, and $v_n=Pv_{n-1}+v_{n-2}$ for all $n\geq 2$. In particular, our result implies that none of these Lucas sequences are stable modulo $3$ either.