Conjecture of Pruthviraj Hajari

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Conjecture of Pruthviraj Hajari

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

$t(x)=\left\{\begin{array}{cl} 8x & \mbox{ if $x$ is odd,}\\ \left\lfloor x/3\right\rfloor & \mbox{ if $x$ is even,} \end{array} \right.$ where

$\lfloor\quad\rfloor$ denotes the floor function.

Equivalently

$t(x)=\left\{\begin{array}{cl} 8x & \mbox{ if $x$ is odd,}\\ x/3 & \mbox{ if $x\equiv 0\pmod{6}$,}\\ (x-2)/3 & \mbox{ if $x\equiv 2\pmod{6}$,}\\ (x-1)/3 & \mbox{ if $x\equiv 4\pmod{6}$.} \end{array} \right.$ All trajectories are conjectured by Pruthviraj Hajari to eventually reach 0 if x ≥ 0. (See MathOverflow question.)

It seems certain that if x < 0, then tⁿ(x) will eventually reach one of the cycles -3 → -24 → -8 → -3, or -5 → -40 → -14 → -5 → -3.

Last modified 10th December 2020
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