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

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 $m$.