The Maximum Length for Ducci Sequences on $Z_m^n$ when $n$ is Even

Abstract

Let $D:{\mathbb{Z}}_m^n\to {\mathbb{Z}}_m^n$ be defined so \[D(x_1, x_2, ..., x_n)=(x_1+x_2 \; \text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m).\] $D$ is known as the Ducci function and for $\mathbf{u}\in {\mathbb{Z}}_m^n$, $\{D^{\hat{I}}\pm (\mathbf{u}){\}}_{\hat{I}\pm =0}^{\infty }$ is the Ducci sequence of $\mathbf{u}$. Every Ducci sequence enters a cycle because ${\mathbb{Z}}_m^n$ is finite. In this paper, we aim to establish an upper bound for how long it will take for a Ducci sequence in ${\mathbb{Z}}_m^n$ to enter its cycle when $n$ is even.