Values of Ducci Periods for Sequences on ${\mathbb{Z}}_m^n$

Abstract

Let $D:{\mathbb{Z}}_m^n\to {\mathbb{Z}}_m^n$ be defined so that \[D(x_1, x_2, ..., x_n)=(x_1+x_2 \; \text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m).\] We call $D$ the Ducci function and the sequence $\{D^{\alpha }(\mathbf{u}){\}}_{\alpha =0}^{\infty }$ the Ducci sequence of $\mathbf{u}$ for $\mathbf{u}\in {\mathbb{Z}}_m^n$. Every Ducci sequence enters a cycle, so we can let $\text{Per}(\mathbf{u})$ be the number of tuples in the Ducci cycle of $\mathbf{u}$, or the period of $\mathbf{u}$. In this paper, we will look at what different possible values of $\text{Per}(\mathbf{u})$ we can have and some conditions that if $\mathbf{u}$ meets at least one of them, $\mathbf{u}$ will generate a period smaller than the maximum period.