A 6-branched generalized-Collatz Conjecture of G. Venturini
Consider the function T:
\[
T(x)=\left\{\begin{array}{ccc}
\frac{x}{6} &\mbox{if $x ≡ 0 \pmod{6}$}\\
\frac{2x+16}{3} &\mbox{if $x ≡ 1 \pmod{6}$}\\
3x+11 &\mbox{if $x ≡ 2 \pmod{6}$}\\
\frac{x-3}{6} &\mbox{if $x ≡ 3 \pmod{6}$}\\
x-4 &\mbox{if $x ≡ 4 \pmod{6}$}\\
\frac{x+9}{2} &\mbox{if $x ≡ 5 \pmod{6}$}.
\end{array}
\right.
\]
It is implicit in Example 5 of G. Venturini that trajectories seem certain to either enter the congruence class B(2,12), or eventually reach one of the two cycles \(0\to 0\) and \(1\to 6\to 1\).
Last modified 31st March 2023
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