Conjecture of Pruthviraj Hajari

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The iterates x, t(x), t(t(x)),... of the mapping

$t(x)=\left\{\begin{array}{cl} 5x+1 & \mbox{ if $x$ is odd,}\\ \left\lceil x/4\right\rceil & \mbox{ if $x$ is even,} \end{array} \right.$ where

$\lceil\quad\rceil$ denotes the ceiling function ( equivalently

$t(x)=\left\{\begin{array}{cl} 5x+1 & \mbox{ if $x$ is odd,}\\ x/4 & \mbox{ if $x\equiv0\pmod{4}$,}\\ (x+2)/4 & \mbox{ if $x\equiv2\pmod{4}$)} \end{array} \right.$ are conjectured by Pruthviraj Hajari to eventually reach 1 if x > 0. (See MathOverflow question.)

It seems certain that if x ≠ 0, then tⁿ(x) will eventually reach one of the cycles 0 → 0, -1 → -4 → -1, or 1 → 6 → 2 → 1, or -3 → -14 → -3.

Last modified 11th December 2020
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