A 3-branched generalized-Collatz Conjecture
Consider the function T1:
\[
T_1(x)=\left\{\begin{array}{cl}
2x & \mbox{ if $3$ divides $x$,}\\
(7x+2)/3 & \mbox{ if $3$ divides $x-1$,}\\
(x-2)/3 & \mbox{ if $3$ divides $x+1$.}
\end{array}
\right.
\]
Clearly T1(x) ≥ 0 if x ≥ 1 and T1(x) < 0 if x < 0.
It is conjectured by Keith Matthews that
- every trajectory starting from x ≥ 1 will eventually enter the zero residue class (mod 3);
- every trajectory starting from x ≤ -1 will eventually enter the zero residue class (mod 3),
or reach one of the cycles -1,-1 or -2,-4,-2.
Similarly, for the function
\[
T_2(x)=\left\{\begin{array}{cl}
2x & \mbox{ if $3$ divides $x$,}\\
(5x-2)/3 & \mbox{ if $3$ divides $x-1$,}\\
(x-2)/3 & \mbox{ if $3$ divides $x+1$,}
\end{array}
\right.
\]
we conjecture that
- every trajectory starting from x ≥ 1 will eventually enter the zero residue class (mod 3), or reach the cycle 1,1;
- every trajectory starting from x ≤ -1 will eventually enter the zero residue class (mod 3),
or reach one of the cycles -1,-1 or -2,-4,-2.
(See article on generalized 3x+1 mappings.)
A $100 Australian prize for a resolution of one of these conjectures is offered.
The algorithms are performed with starting values y= 3M+1 and y=3M-1.
The iterates y, Ti(y), Ti(Ti(y)),... of each function are printed and the number of steps taken to reach either a multiple of 3, or else one of the cycles, is recorded.
Last modified 4th December 2013
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