فهرست نشریات علمی دسترسی آزاد ایران

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 $\{(x_1, x_2,\ldots,x_n) \in \mathbb{Z}_m^n \; \mid \; x_1+x_2+ \cdots +x_n \equiv 0 \; \text{mod} \; 2^l\}$.