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Let \(\Omega \) be a set of positive integers and let \(f:\Omega \rightarrow \Omega \) be an arithmetic function. Let \(V = (v_i)_{i=1}^n\) be a finite sequence of positive integers. An integer \(m \in \Omega \) has 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 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 f. This paper gives a proof that the Syracuse function is wildly increasing-decreasing.

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Increasing-Decreasing Patterns in the Iteration of an Arithmetic Function

  • 会议论文

  • 发表日期：

    2025年

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  • Melvyn B. Nathanson

摘要

Let \(\Omega \) be a set of positive integers and let \(f:\Omega \rightarrow \Omega \) be an arithmetic function. Let \(V = (v_i)_{i=1}^n\) be a finite sequence of positive integers. An integer \(m \in \Omega \) has 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 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 f. This paper gives a proof that the Syracuse function is wildly increasing-decreasing.