Increasing-decreasing patterns in the iteration of an arithmetic function
Melvyn B. Nathanson
TL;DR
The paper investigates whether the Syracuse function $S(m)$ can realize every prescribed alternating pattern of increases and decreases for odd positive integers. It develops a matrix-based realization: a built matrix $M$ with carefully chosen integer sequences yields a surjective map from $\mathbb{Z}^{n+1}$ to $\mathbb{Z}^n$, guaranteeing the existence of odd-positive solutions to $M\mathbf x=\mathbf h$ and hence the desired pattern $V$ via a finite Diophantine system; with two proofs provided (diophantine construction and Smith normal form). A direct Syracuse-specific analysis furnishes explicit monotone runs, and a modular, simpler proof shows that for any pattern sequence $\Lambda$ there exists a modulus $2^{\ell}$ and residue $r$ such that $m\equiv r\pmod{2^{\ell}}$ yields the pattern, establishing wild increasing-decreasing without solving the full Diophantine system. The work demonstrates a constructive framework for pattern realizability in discrete dynamical systems and raises open questions about related aliquot-like trajectories and necessary-sufficient conditions for wild behavior.
Abstract
Let $Ω$ be a set of positive integers and let $f:Ω\rightarrow Ω$ be an arithmetic function. Let $V = (v_i)_{i=1}^n$ be a finite sequence of positive integers. An integer $m \in Ω$ has \textit{increasing-decreasing pattern} $V$ with respect to $f$ if, for all odd integers $i \in \{1,\ldots, n\}$, \[ f^{v_1+ \cdots + v_{i-1}}(m) < f^{v_1+ \cdots + v_{i-1}+1}(m) < \cdots < f^{v_1+ \cdots + v_{i-1}+v_{i}}(m) \] and, for all even integers $i \in \{2,\ldots, n\}$, \[ f^{v_1+ \cdots + v_{i-1}}(m) > f^{v_1+ \cdots +v_{i-1}+1}(m) > \cdots > f^{v_1+ \cdots +v_{i-1}+v_i}(m). \] The arithmetic function $f$ is \textit{wildly increasing-decreasing} if, for every finite sequence $V$ of positive integers, there exists an integer $m \in Ω$ such that $m$ has increasing-decreasing pattern $V$ with respect to $f$. This paper gives a proof that the Syracuse function is wildly increasing-decreasing.