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 \(1\to 6\to 1\).

2+12k → 17+36k → 13+18k → 14+12k → 53+36k → 31+18k → ⋯

Once the trajectory arrives in B(2,12), it is composed of three arithmetic progressions:

Enter y:
Last modified 19st May 2026
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