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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2410.18204 |
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| _version_ | 1866909361667833856 |
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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 \[D(x_1, x_2, ..., x_n)=(x_1+x_2 \; \text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m).\] $D$ is known as the Ducci function and for $\mathbf{u} \in \mathbb{Z}_m^n$, $\{D^α(\mathbf{u})\}_{α=0}^{\infty}$ is the Ducci sequence of $\mathbf{u}$. Every Ducci sequence enters a cycle because $\mathbb{Z}_m^n$ is finite. In this paper, we aim to establish an upper bound for how long it will take for a Ducci sequence in $\mathbb{Z}_m^n$ to enter its cycle when $n$ is even. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_18204 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Maximum Length for Ducci Sequences on $\mathbb{}Z_m^n$ when $n$ is Even 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 \[D(x_1, x_2, ..., x_n)=(x_1+x_2 \; \text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m).\] $D$ is known as the Ducci function and for $\mathbf{u} \in \mathbb{Z}_m^n$, $\{D^α(\mathbf{u})\}_{α=0}^{\infty}$ is the Ducci sequence of $\mathbf{u}$. Every Ducci sequence enters a cycle because $\mathbb{Z}_m^n$ is finite. In this paper, we aim to establish an upper bound for how long it will take for a Ducci sequence in $\mathbb{Z}_m^n$ to enter its cycle when $n$ is even. |
| title | The Maximum Length for Ducci Sequences on $\mathbb{}Z_m^n$ when $n$ is Even |
| topic | Number Theory Group Theory 20D60, 11B83, 11B50 |
| url | https://arxiv.org/abs/2410.18204 |