### A generalization of the 3x+1 mapping due to Geoffrey C. Caveney

In a posting to sci.math on February 1, 1998, Geoffrey Caveney defined a mapping f_{a,b}: ℤ → ℤ as follows: Let a > b > 1, gcd(a,b) = 1. Then

For example, a = 3, b = 2 gives the 3x+1 mapping:

Caveney was not aware that his mapping was a special case of a mapping studied by Herbert Möller in 1978, which in turn was generalized in 1984 to the case of a *relatively prime* mapping by K.R. Matthews and A.M. Watts.
Caveney's mapping can also be written as a b-branched mapping:

This reduces to W. Carnielli's mapping if a = b+1, as r_{i} = b - i if 1 ≤ i < b.
According to Möller, it seems certain that for x ∈ ℤ, the sequence of iterates x, T(x), T^{2}(x), ...
always eventually enters a cycle if a < b^{b/(b-1)} and that regardless of this inequality, there are only finitely many such cycles. Also if a > b^{b/(b-1)}, the prediction is that most trajectories will be divergent.

We note that writing a=b+x, x ≥ 1, we have log(1+x/b) < x/b, (1+x/b)^{b} < e^{x}, so if x + b > e^{x}, we have (1+x/b)^{b} < b+x and hence (b+x)^{b-1} < b^{b}.

Hence a < b^{b/(b-1)} if b > e^{a-b} -(a-b). In particular, we have the following inequalities:

- b+1 < b
^{b/(b-1)} iff b ≥ 2.
- b+2 < b
^{b/(b-1)} iff b ≥ 3.
- b+3 < b
^{b/(b-1)} iff b ≥ 12.
- b+4 < b
^{b/(b-1)} iff b ≥ 42.

Caveny experimented with a/b: 4/3, 5/3, 5/4, 6/5, 7/5, 8/5 and 7/6. It seems that for f_{7,5}, where

all iterates will eventually enter one of the following cycles:
- 0,0
- -1,-1
- 1,2,3,5,1
- -2,-2
- 6,9,13,19,27,38,54,76,107,150,30,6
- -17,-23,-32,-44,-61,-85,-17
- -33,-46,...,-165,-33 (length 58)

It seems that for f_{5,3}, where

all iterates will eventually enter one of the following cycles:
- 0,0
- -1,-1
- 4,7,12,4
- 8,14,24,8.

For f_{8,5}, we have found five cycles:
- 0,0
- -1,-1
- 4,7,1,2,20,4
- -11,-17,-27,-43,-68,-108,-172,-275,-55,-11
- -49,-78,...,-245,-49 (length 62)

Here a = 8 > b^{b/(b-1)} = 5^{5/4} = 7.476···. It seems certain that the trajectories starting with 8 and -98 diverge.

The viewer can experiment with f_{5,3} and f_{7,5}:

*Last modified 22nd August 2011*

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