### A generalization of the 3x-1 mapping due to Lu Pei

Consider the mapping T_{d}: ℤ → ℤ
Let d ≥ 2. Then

T_{d}(n) = n/d if n ≡ 0 (mod d)
T_{d}(n) = ((d+1)n - i)/d if n ≡ i (mod d), -d/2 < i ≤ d/2, i ≠ 0.

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 is 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.
Clearly T_{d}(n) = n for -d/2 < n ≤ d/2.

For d = 3, 6 and 10, we appear to get no other cycles.

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 assume that every trajectory will eventually reach a cycle and use Floyd's method of testing for equality of k^{th} and 2k^{th} iterates for find a cycle element. We list the cycle with smallest absolute value as starting point.

*Last modified 21st July 2011*

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