are printed and the number of steps taken to reach one of the integers 0, -1, -4, -19 is recorded.

Benoit Cloitre in an email to Jeff Lagarias dated July 19, 2011, has conjectured that every trajectory starting from
a positive integer will eventually reach 0.

It is also clear experimentally that any trajectory starting from a negative integer will eventually reach one of three cycles beginning with -1, -4 or -19.

Cycles: (i) 0,0; (ii) -1,-2,-1; (iii) -4,-17,-6,-4; (iv) -19,-92,-31,-152,-51,-34,-167,-56,-19.

Heuristics: Visit *Generalized 3x+1 functions and Markov matrices*, where we find,
with d = 3 and m_{0}=2, m_{1} = 1, m_{2} = 15 and x_{0}=0, x_{1} = 0, x_{2} = 3,

Q(3) = 1/3 1/3 1/3 1/3 1/3 1/3 0 1 0has stationary vector (1/4, 1/2, 1/4). Also

(2/3)^{1/4}(1/3)^{1/2}(15/3)^{1/4} < 1.

Consequently we expect all trajectories to eventually cycle.

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Benoit Cloitre in an email to Jeff Lagarias dated July 20, 2011, has conjectured that every trajectory starting from a positive integer will eventually reach 0 or 2.It is also clear experimentally that any trajectory starting from a negative integer will eventually reach -1.

Cycles: (i) 0,0; (ii) 2,7,2: (iii) -1,-1.

Heuristics: This function can be regarded as a six-branched mapping. Accordingly, visit *Generalized 3x+1 functions and Markov matrices*, where we find,

with d = 6 and m_{0} = 21, m_{1} = 2, m_{2} = 21, m_{3} = 2, m_{4} = 21, m_{5} = 2
and x_{0} = 0 = x_{1} = x_{2} = x_{3} = x_{4} = x_{5}, that

Q(6) = 1/2 0 0 1/2 0 0 1/3 0 1/3 0 1/3 0 0 1/2 0 0 1/2 0 0 1/3 0 1/3 0 1/3 0 0 1/2 0 0 1/2 0 1/3 0 1/3 0 1/3This Markov matrix has stationary vector (2/15,3/15,2/15,3/15,2/15,3/15) and the weighted product (21/6)

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For this mapping, Benoit Cloitre has conjectured that every trajectory starting from a positive integer will eventually reach 0, 4, 6 or 24.It is also clear experimentally that any trajectory starting from a negative integer will eventually reach -1.

Cycles: (i) 0,0; (ii) 4,13,4: (iii) 6,19,6 (iv) 24, 78,...,73,24 (length 87) (v) -1,-1.

Heuristics: This function can be regarded as a twelve-branched mapping and an analysis of the 12x12 Markov matrix Q(12) suggests that all trajectories will eventually cycle.

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For m = 3, 5 and m ≥ 7, Benoit Cloitre conjectures that the trajectories of FThis mapping can be regarded as a 2m-branched generalized 3x+1 mapping if m is even, and an m-branched mapping if m is odd. See the BC program cloitrem in which

Here all trajectories appear to enter one of six cycles (i) 0,0 (ii) -1,-1 (iii) 1,2,1 (iv) -2,-3,-4,-2 (v) -5,-10,-7 (vi) 19,38,25,50,33,22,44,29,19.

We can use the Markov matrix approach of *generalized 3x+1 functions* to predict the behaviour of trajectories. See manuscript.

We find for example, with m = 3, m_{0}=2, m_{1} = 6, m_{2} = 2 and x_{0} = 0= x_{1} = x_{2},

Q(3) = 1/3 1/3 1/3 0 0 1 1/3 1/3 1/3which has stationary vector (1/4, 1/4, 1/2). Also

(2/3)^{1/4}(6/3)^{1/4}(2/3)^{1/2} < 1.

Consequently we expect all trajectories to eventually cycle.

A list of cycles found for 2 ≤ m ≤ 2000 is also attached.

The examples of F_{29} and F_{153} are striking, as there are apparently 165 and 416 cycles, respectively.

*Last modified 12th August 2011*

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