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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2403.05319 |
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| _version_ | 1866913485257965568 |
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| author | Lewis, Mark L. Tefft, Shannon M. |
| author_facet | Lewis, Mark L. Tefft, Shannon M. |
| contents | 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 $\{D^α(\mathbf{u})\}_{α=0}^{\infty}$ 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 $\{(x_1, x_2, ..., x_n) \in \mathbb{Z}_m^n \; \mid \; x_1+x_2+ \cdots +x_n \equiv 0 \; \text{mod} \; 2^l\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_05319 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ducci on $\mathbb{Z}_m^n$ and the Maximum Length for $n$ Odd Lewis, Mark L. Tefft, Shannon M. Number Theory Group Theory 20D60, 11B83, 11B50 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 $\{D^α(\mathbf{u})\}_{α=0}^{\infty}$ 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 $\{(x_1, x_2, ..., x_n) \in \mathbb{Z}_m^n \; \mid \; x_1+x_2+ \cdots +x_n \equiv 0 \; \text{mod} \; 2^l\}$. |
| title | Ducci on $\mathbb{Z}_m^n$ and the Maximum Length for $n$ Odd |
| topic | Number Theory Group Theory 20D60, 11B83, 11B50 |
| url | https://arxiv.org/abs/2403.05319 |