Ducci on $\mathbb{Z}_m^n$ and the Maximum Length for $n$ Odd

Abstract

Define the Ducci function $D: \mathbb{Z}_m^n \to \mathbb{Z}_m^n$ so \[D(x_1,x_2, ...,x_n)=(x_1+x_2 \;\text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m).\] Call $\{D^{\alpha}(\mathbf{u})\}_{\alpha=0}^{\infty}$ the Ducci sequence of $\mathbf{u}$. Because $\mathbb{Z}_m^n$ is finite, every Ducci sequence will enter a cycle. In this paper, we will prove that if $n$ is odd and $m=2^lm_1$ where $m_1$ is odd, then the longest it will take for a Ducci sequence to enter its cycle is $l$ iterations. Furthermore, we will prove the set of all tuples in a cycle for $\mathbb{Z}_m^n$ is $\{(x_1, x_2, ..., x_n) \in \mathbb{Z}_m^n \; \mid \; x_1+x_2+ \cdots +x_n \equiv 0 \; \text{mod} \; 2^l\}$.