Let $\Omega$ be a set of positive integers and let $S : \Omega\to\Omega$ be an arithmetic function. Let $V = (v_i)^n_{i=1}$ be a finite sequence of positive integers. An integer $m\in\Omega$ has increasing-decreasing pattern $V$ with respect to $S$ if, for odd integers $i\in \{1, \ldots , n\}$, we have $$S^{v_1+\ldots+v_{i−1}} (m) \lt S^{v_1+\ldots+v_{i−1}+1}(m) \lt \cdots \lt S^{v_1+\ldots+v_{i−1}+v_i (m)}$$ and, for even integers $i\in \{1, \ldots , n\}$, we have $$S^{v_1+\ldots+v_{i−1}} (m) \gt S^{v_1+\ldots+v_{i−1}+1}(m) \gt \cdots \gt S^{v_1+\ldots+v_{i−1}+v_i (m)}.$$ The arithmetic function $S$ is wildly increasing-decreasing if, for every finite sequence $V$ of positive integers, there exists an integer $m\in\Omega$ such that $m$ has increasing-decreasing pattern $V$ with respect to $S$. This talk describes a new proof that the Collatz function is wildly increasing-decreasing.
Mel Nathanson