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 $m=2^l{m}_1$ 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 $\{(x_1,x_2,...,x_n)\in {\mathbb{Z}}_m^n\mid x_1+x_2+\cdots +x_n\equiv 0\text{mod}{2}^l\}$.