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

TL;DR

This paper extends known results about the periodic residues of Fibonacci sequences modulo powers of 3 to a broader class of Lucas sequences, showing they also lack stability modulo 3.

Contribution

It generalizes the frequency and residue distribution results from Fibonacci to various Lucas sequences modulo powers of 3.

Findings

Fibonacci sequence has a period length of 4·3^{k-1} modulo 3^k.

Lucas sequences defined by P also have specific residue distributions.

All these sequences are not stable modulo 3.

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.