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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2401.17502 |
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| _version_ | 1866913485210779648 |
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| author | Lewis, Mark L. Tefft, Shannon M. |
| author_facet | Lewis, Mark L. Tefft, Shannon M. |
| contents | Let the Ducci function $D: \mathbb{Z}_m^n \to \mathbb{Z}_m^n$ be defined as
\[D(x_1, x_2, ..., x_n)=(x_1+x_2 \; \text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m)\]
and let the Ducci sequence of $\mathbf{u}$ be the sequence $\{D^α(\mathbf{u})\}_{α=0}^{\infty}$.
%In this paper, we will prove that if $n,m$ are powers of $2$, then repeatedly applying $D$ will eventually result in $(0,0,...,0)$, as well as establish an upper bound for how many iterations it will take for this to happen.
In this paper, we will provide another proof that for $n=2^k$ and $m=2^l$, that all Ducci sequences will end in $(0,0,...,0)$ and additionally prove that this will happen in at most $2^{k-1}(l+1)$ iterations of $D$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_17502 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Period of Ducci Cycles on $\mathbb{Z}_{2^l}$ for Tuples of Length $2^k$ Lewis, Mark L. Tefft, Shannon M. Number Theory Group Theory 20D60, 11B83, 11B50 Let the Ducci function $D: \mathbb{Z}_m^n \to \mathbb{Z}_m^n$ be defined as \[D(x_1, x_2, ..., x_n)=(x_1+x_2 \; \text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m)\] and let the Ducci sequence of $\mathbf{u}$ be the sequence $\{D^α(\mathbf{u})\}_{α=0}^{\infty}$. %In this paper, we will prove that if $n,m$ are powers of $2$, then repeatedly applying $D$ will eventually result in $(0,0,...,0)$, as well as establish an upper bound for how many iterations it will take for this to happen. In this paper, we will provide another proof that for $n=2^k$ and $m=2^l$, that all Ducci sequences will end in $(0,0,...,0)$ and additionally prove that this will happen in at most $2^{k-1}(l+1)$ iterations of $D$. |
| title | The Period of Ducci Cycles on $\mathbb{Z}_{2^l}$ for Tuples of Length $2^k$ |
| topic | Number Theory Group Theory 20D60, 11B83, 11B50 |
| url | https://arxiv.org/abs/2401.17502 |