Ducci on $\mathbb{Z}_m^n$ and the maximum length for $n$ odd

Lewis, Mark L.; Tefft, Shannon M.

doi:10.22108/ijgt.2025.142851.1926

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	Ducci on $\mathbb{Z}_m^n$ and the maximum length for $n$ odd
International Journal of Group Theory
مقالات آماده انتشار، اصلاح شده برای چاپ، انتشار آنلاین از تاریخ 30 اردیبهشت 1404 اصل مقاله (473.96 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/ijgt.2025.142851.1926
نویسندگان
Mark L. Lewis؛ Shannon M. Tefft^*
Department of Mathematical Sciences, Kent State University, Kent, OH, United States
چکیده
Let the Ducci function, $D$, be an endomorphism on $\mathbb{Z}_m^n$ such that \[D(x_1, x_2,\ldots,x_n)=(x_1+x_2 \;\text{mod} \; m, x_2+x_3 \; \text{mod} \; m,\ldots, x_n+x_1 \; \text{mod} \; m).\] The sequence $\{D^{\alpha}(\mathbf{u})\}_{\alpha=0}^{\infty}$ is the Ducci sequence of $\mathbf{u}$ for $\mathbf{u} \in \mathbb{Z}_m^n$. Because $\mathbb{Z}_m^n$ is finite, the Ducci sequence of $\mathbf{u}$ enters a cycle for all $\mathbf{u} \in \mathbb{Z}_m^n$, which we call the Ducci cycle of $\mathbf{u}$. In this paper, our main goal is to 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 in $\mathbb{Z}_m^n$ to enter its cycle is $l$ iterations of $D$. In addition to this, we will prove that the set of all tuples in $\mathbb{Z}_m^n$ in a Ducci cycle for some $\mathbf{u} \in \mathbb{Z}_m^n$ is $\{(x_1, x_2,\ldots,x_n) \in \mathbb{Z}_m^n \; \mid \; x_1+x_2+ \cdots +x_n \equiv 0 \; \text{mod} \; 2^l\}$.
کلیدواژه‌ها
Ducci sequence؛ modular arithmetic؛ period؛ $n$-Number Game

مراجع
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Ducci on $\mathbb{Z}_m^n$ and the maximum length for $n$ odd