Ducci on $\mathbb{Z}_m^3$ and the Max Period

Mark L. Lewis; Shannon M Tefft

Ducci on $\mathbb{Z}_m^3$ and the Max Period

Number Theory 2024-03-15 v2 Group Theory

Authors: Mark L. Lewis , Shannon M Tefft

Abstract

Let $D(x_1, x_2, ..., x_n)=(x_1+x_2 \;\text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m)$ where $D \in End(\mathbb{Z}_m^n)$ be the Ducci function. The sequence $\{D^k(\mathbf{u})\}_{k=0}^{\infty}$ will eventually enter a cycle. If $n=3$ , we aim to establish the longest a cycle can be for a given $m$ .

Keywords

Ducci sequence modular arithmetic cycle length finite rings dynamical systems number theory periodicity

Cite

@article{arxiv.2402.07833,
  title  = {Ducci on $\mathbb{Z}_m^3$ and the Max Period},
  author = {Mark L. Lewis and Shannon M Tefft},
  journal= {arXiv preprint arXiv:2402.07833},
  year   = {2024}
}

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