### A 3-branched generalized-Collatz Conjecture

Consider the function T_{1}:

T_{1}(x) | = | 2x | if 3 divides x, |

T_{1}(x) | = | (7x+2)/3 | if 3 divides x-1, |

T_{1}(x) | = | (x-2)/3 | if 3 divides x+1. |

Clearly T_{1}(x) ≥ 0 if x ≥ 1 and T_{1}(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) | = | 2x | if 3 divides x, |

T_{2}(x) | = | (5x-2)/3 | if 3 divides x-1, |

T_{2}(x) | = | (x-2)/3 | if 3 divides x+1, |

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, T_{i}(y), T_{i}(T_{i}(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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