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 $m$.

Figures (2)

• Figure 1: Transition Graph for $\mathbb{Z}_6^3$
• Figure 2: Transition Graph for $\mathbb{Z}_3^3$

Theorems & Definitions (24)

• Definition 1
• Theorem 2
• Theorem 3
• proof
• Theorem 4
• proof
• Lemma 5
• proof
• Theorem 6
• proof