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SUMMARY:Mel Nathanson (Lehman College (CUNY))
DTSTART:20230526T203000Z
DTEND:20230526T205500Z
DTSTAMP:20260423T024835Z
UID:CANT2023/65
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2023/65/
 ">Patterns in the iteration of an arithmetic function</a>\nby Mel Nathanso
 n (Lehman College (CUNY)) as part of Combinatorial and additive number the
 ory (CANT 2023)\n\nLecture held in Room 4102 in CUNY Grad Center.\n\nAbstr
 act\nLet $\\Omega$ be a set of positive integers and let $S:\\Omega \\righ
 tarrow \\Omega$\n be an arithmetic function.  Let $V = (v_i)_{i=1}^n$ be a
  finite sequence of positive integers.  \nAn integer $m \\in \\Omega$ has 
 \\textit{increasing-decreasing pattern} $V$ with respect to $S$ if\,  \nfo
 r all odd integers $i \\in \\{1\,\\ldots\, n\\}$\,  \n\\[\nS^{v_1+ \\cdots
  + v_{i-1}}(m) < S^{v_1+ \\cdots + v_{i-1}+1}(m) < \\cdots < S^{v_1+ \\cdo
 ts + v_{i-1}+v_{i}}(m)\n\\]\nand\, for all even  integers $i \\in \\{2\,\\
 ldots\, n\\}$\, \n\\[\nS^{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).\n\\]\nThe arit
 hmetic function $S$ is \\textit{wildly increasing-decreasing} if\,  \nfor 
 every finite sequence $V$ of positive integers\, there exists an integer $
 m  \\in \\Omega$ \nsuch that $m$ has increasing-decreasing pattern $V$ wit
 h respect to $S$.  \nIt is proved that the Collatz function \nis wildly in
 creasing-decreasing.\n
LOCATION:https://researchseminars.org/talk/CANT2023/65/
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