Values of Ducci Periods for Sequences on $\mathbb{Z}_m^n$
Mark L. Lewis, Shannon M. Tefft
TL;DR
This work classifies the possible periods of Ducci sequences on the finite group $Z_m^n$ by analyzing when the maximum period $P_m(n)$ is attained and when smaller cycles arise. It introduces the coefficient framework $a_{r,s}$ to track Ducci iterates, specializes to $n=3$ with $a_r,b_r,c_r$, and proves key parity results such as $6|P_m(3)$ for $m>2$, which underpin the prime-modulus analysis. For prime moduli, most non-sum, non-uniform triples achieve the maximum period, with a practical linear-system method to rule out nonmaximal divisors of $P_m(n)$; the paper also details exceptional cases and symmetry structures. In the special case $n=m=p$ prime, it proves that only three period lengths occur: $1$, the order of $2$ modulo $p$ for uniform vectors, and the maximum $p\delta$, with a determinant-based argument enforcing the dichotomy. Overall, the results illuminate the algebraic/combinatorial structure of Ducci dynamics on $Z_m^n$ and furnish methods to determine achievable periods, highlighting when smaller cycles inevitably appear.
Abstract
Let $D: \mathbb{Z}_m^n \to \mathbb{Z}_m^n$ be defined so that \[D(x_1, x_2, ..., x_n)=(x_1+x_2 \; \text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m).\] We call $D$ the Ducci function and the sequence $\{D^α(\mathbf{u})\}_{α=0}^{\infty}$ the Ducci sequence of $\mathbf{u}$ for $\mathbf{u} \in \mathbb{Z}_m^n$. Every Ducci sequence enters a cycle, so we can let $\text{Per}(\mathbf{u})$ be the number of tuples in the Ducci cycle of $\mathbf{u}$, or the period of $\mathbf{u}$. In this paper, we will look at what different possible values of $\text{Per}(\mathbf{u})$ we can have and some conditions that if $\mathbf{u}$ meets at least one of them, $\mathbf{u}$ will generate a period smaller than the maximum period.