### Walter Carnielli's generalization of the 3x+1 mapping

The following mapping T_{d}: ℤ → ℤ was defined in an email from Professor
Walter Carnielli to Keith Matthews (26th December 2010).

Let d ≥ 2. Then

T_{d}(n) = n/d if n ≡ 0 (mod d)
T_{d}(n) = ((d+1)n + d - i)/d if n ≡ i (mod d), 1 ≤ i ≤ d-1.

For example, d = 2 gives the 3x+1 mapping:
T_{2}(n) = n/2 if n ≡ 0 (mod 2)
T_{2}(n) = (3n + 1)/2 if n ≡ 1 (mod 2).

This is a special case of a version of a mapping studied by Herbert Möller
and also an example of a *relatively prime* mapping, in the language of Matthews and Watts, where m_{0}=1 and m_{i} = d+1 for 1 ≤ i < d and where we have the inequality
m_{0}m_{1}⋯m_{d} =(d+1)^{d-1} < d^{d}.

So it seems certain that the sequence of iterates
n, T_{d}(n), T_{d}^{2}(n), ...

always eventually enters a cycle and that there are only finitely many such cycles as was conjectured by Professor Carnielli.
It is easy to prove that

- T
_{d}(n) = n for n = -1,...,-(d – 1);
- 1, 2,..., d is a cycle.

For d = 7,14, 18 and 21, we appear to get no cycles other than those in (i) and (ii).

It would be interesting to determine all d with this property. See the Table, which was constructed using the author's faster CALC program.
We search all trajectories x, T_{d}(x), T_{d}^{2}(x),... where |x| ≤
R ⁄ 2 = 60000 ⁄ 2 for cycling by Floyd's method of testing for equality of k^{th} and 2k^{th} iterates for k ≤ U = 1000000.

Here we list cycles found other than those of cases (i) and (ii) for all d satisfying m ≤ d ≤ n, where m and n satisfy
2 ≤ m ≤ n ≤ 100.

We choose the cycle element with smallest absolute value as starting point.

Finally, we limit the number of cycles for each d to 500.

*Last modified 17th January 2011*

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