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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.03348 |
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Table of Contents:
- 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.