Bonded Trajectories of the $3x+\gamma$ Problem

Abstract

Fix $\gamma \in \mathbb{Z}_{>0}^{odd}$ and $n\in\mathbb{Z}_{>0}$. We define the function $C_\gamma:{\mathbb Z}_{>0}\to {\mathbb Z}_{>0}$ such that if $n$ is odd, $C_\gamma(n)=3n+\gamma$; and if $n$ is even, $C_\gamma(n)=n/2$. We define the characteristic mapping $\chi_\gamma: {\mathbb Z}_{>0}\to \{0,\, 1\}$ to be $\chi_\gamma(n)\equiv C_\gamma(n)\, {\rm mod}\, 2$. Let $n$ start an integral loop of length $N$ associated with the $3x+\gamma$ Problem. Let $\rho$ and $\nu$ be the count of the number of zeros and ones in a single period of $B = \left(\chi_\gamma^i(n)\right)_{i\geq 0}$. In a single period of $B$, let $m_j$ denote the number of zeros between the $(j-1)$th and $j$th one. Let $\mathcal{M}_n$ be the matrix associated to $n$ whose elements are the sequential products of $2^{m_j}$ (e.g. $(2^{m_0},2^{m_0+m_1},2^{m_0+m_1+m_2},...)$). Let $p$ be a prime factor for all the terms in the integral loop starting with $n$ with multiplicity $a>0$. Suppose also that $p$ is a prime factor of $\gamma$ and $2^\rho - 3^\nu$ with multiplicity $b$ and $c$, respectively. Finally assume that $c>b-a$. Then $\det(\mathcal{M}_n) \equiv 0 \pmod p$. We do find examples of this property. Let $\nu$ be prime. Let $z_j = 2^{(m_j+2m_{j+1}+3m_{j+2}+...)/\nu}$ be a weighted arithmetic average of the $m_j$. We prove that if $p$ is a prime factor of $2^\rho-3^\nu$ distinct from $\nu$ so that the residue class $p \pmod \nu$ generates the whole group $\mathbb{Z}_\nu^{*}$ then $p\nmid \det(\mathcal{M}_n)$ if and only if $p\nmid(z_1+...+z_\nu)$ and for any $1\leq i<j\leq \nu$ we have $z_i \not\equiv z_j \pmod p$. By this, we give an interesting property for the integral loop.