Patterns in the iteration of an arithmetic function

Melvyn B. Nathanson

Published 2022-08-03Version 1

Let $S$ be an arithmetic function and let $V = (v_i)_{i=1}^n$ be a finite sequence of positive integers. An integer $m$ has increasing-decreasing pattern $V$ with respect to $S$ if, for odd $i$, \[ S^{v_1+ \cdots + v_{i-1}}(m) < S^{v_1+ \cdots + v_{i-1}+1}(m) < \cdots < S^{v_1+ \cdots + v_{i-1}+v_{i}}(m) \] and, for even $i$, \[ S^{v_1+ \cdots + v_{i-1}}(m) > S^{v_1+ \cdots +v_{i-1}+1}(m) > \cdots > S^{v_1+ \cdots +v_{i-1}+v_i}(m). \] The arithmetic function $S$ is wildly increasing-decreasing if, for every finite sequence $V$ of positive integers, there exists an integer $m$ such that $m$ has increasing-decreasing pattern $V$ with respect to $S$. This paper gives a new proof that the Collatz function is wildly increasing-decreasing.