Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

J. C. Saunders, R. Nicholas Stephens

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.

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

Abstract

The Fibonacci sequence defined by , , and has a shortest period length of modulo for every . In 2011, Bundschuh and Bundschuh \cite{bundschuh3} gave the frequencies of every residue in this shortest period. In particular, their result implies that the Fibonacci sequences is not stable modulo . Here we extend this result to other Lucas sequences. More specifically, we give analogous results for Lucas sequences defined by with , , and for all , as well as Lucas sequences defined by with , , and for all . In particular, our result implies that none of these Lucas sequences are stable modulo either.

Paper Structure

This paper contains 8 sections, 22 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Classical Lemmas on Lucas Sequences
  3. The Case of $3\nmid P$
  4. Preliminary Lemmas
  5. The Proof
  6. The Case of $3\mid P$
  7. Future Work
  8. Acknowledgements

Key Result

Theorem 1

Let $3\nmid P$ and $3^{\delta}\mathrel\Vert P^2+2$. 1) Suppose $\delta=1$ and $k\in\mathbb{N}$. Then for every residue $b$ we have and 2) Suppose $\delta\geq 2$ and $k\in\mathbb{N}$, $k\geq 2\delta-1$. Then for every residue $b$ we have and 3) Suppose $\delta\geq 2$ and $k\in\mathbb{N}$, $\delta\leq k<2\delta-1$. Then for every residue $b$ we have and

Theorems & Definitions (42)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 32 more