A New Proof of Rational Cycles for Collatz-Like Functions Using a Coprime Condition

Abstract

In this paper, we study the bounded trajectories of Collatz-like functions. Fix $\alpha ,\beta \in {\mathbb{Z}}_{>0}$ so that $\alpha$ and $\beta$ are coprime. Let $\overline{k}=\left(k_1,\dots ,k_{\beta -1}\right)$ so that for each $1\le i\le \beta -1$ , $k_i\in {\mathbb{Z}}_{>0}$ , $k_i$ is coprime to $\alpha$ and $\beta$ , and $k_i\equiv i\left(\mathrm{mod}\beta \right)$ . We define the function $C_{\left(\alpha ,\beta ,\overline{k}\right)}:{\mathbb{Z}}_{>0}⟶{\mathbb{Z}}_{>0}$ and the sequence $\left\{n,C_{\left(\alpha ,\beta ,\overline{k}\right)}\left(n\right),C_{\left(\alpha ,\beta ,\overline{k}\right)}^2\left(n\right),\cdots \right\}$ a trajectory of $n$ . We say that the trajectory of $n$ is an integral loop if there exists some $N$ in ${\mathbb{Z}}_{>0}$ so that $C_{\left(\alpha ,\beta ,\overline{k}\right)}^N\left(n\right)=n$ . We define the characteristic mapping ${\chi }_{\left(\alpha ,\beta ,\overline{k}\right)}:{\mathbb{Z}}_{>0}⟶\left\{\mathrm{0,1},\dots ,\beta -1\right\}$ and the sequence $\left\{n,{\chi }_{\left(\alpha ,\beta ,\overline{k}\right)}\left(n\right),{\chi }_{\left(\alpha ,\beta ,\overline{k}\right)}^2\left(n\right),\cdots \right\}$ the characteristic trajectory of $n$ . Let $B\in {\mathbb{Z}}_{\beta }$ be a $\beta$ -adic sequence so that $B={\left({\chi }_{\left(\alpha ,\beta ,\overline{k}\right)}^i\left(n\right)\right)}_{i\ge 0}$ . We say that $B$ is eventually periodic if it eventually has a purely $\beta$ -adic expansion. We show that the trajectory of $n$ eventually enters an integral loop if and only if $B$ is eventually periodic.