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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