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Let $\Omega$ be a set of positive integers and let $S:\Omega \rightarrow\Omega$  be an arithmetic function. Let $V = (v_i)_{i=1}^n$ be a finite sequence ofpositive integers. An integer $m \in \Omega$ has \textit{increasing-decreasingpattern} $V$ with respect to $S$ if, for all odd integers $i \in \{1,\ldots,n\}$, \[ S^{v_1+ \cdots + v_{i-1}}(m) < S^{v_1+ \cdots + v_{i-1}+1}(m) < \cdots< S^{v_1+ \cdots + v_{i-1}+v_{i}}(m) \] and, for all even integers $i \in\{2,\ldots, n\}$, \[ S^{v_1+ \cdots + v_{i-1}}(m) > S^{v_1+ \cdots+v_{i-1}+1}(m) > \cdots > S^{v_1+ \cdots +v_{i-1}+v_i}(m). \] The arithmeticfunction $S$ is \textit{wildly increasing-decreasing} if, for every finitesequence $V$ of positive integers, there exists an integer $m \in \Omega$ suchthat $m$ has increasing-decreasing pattern $V$ with respect to $S$. This papergives a proof that the Syracuse function is wildly increasing-decreasing.

Patterns in the iteration of an arithmetic function