Values of Ducci Periods for Sequences on $\mathbb{Z}_m^n$
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arXiv
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| Format: | Preprint |
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2025
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| author | Lewis, Mark L. Tefft, Shannon M. |
| author_facet | Lewis, Mark L. Tefft, Shannon M. |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_03348 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Values of Ducci Periods for Sequences on $\mathbb{Z}_m^n$ Lewis, Mark L. Tefft, Shannon M. Number Theory Group Theory 20D60, 11B83, 11B50 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. |
| title | Values of Ducci Periods for Sequences on $\mathbb{Z}_m^n$ |
| topic | Number Theory Group Theory 20D60, 11B83, 11B50 |
| url | https://arxiv.org/abs/2502.03348 |