A generalization of the 3x-1 mapping due to Lu Pei
Consider the mapping Td: ℤ → ℤ
Let d ≥ 2. Then
\[
T_d(n)=\left\{\begin{array}{cl}
n/d& \mbox{ if \(n\equiv0\pmod{d}\),}\\
((d+1)n - i)/d& \mbox{ if \(n\equiv i\pmod{d}, -d/2\lt i\le d/2, i\neq0\).}
\end{array}
\right.
\]
For example, d = 2 gives the 3x-1 mapping:
\[
T_2(n)=\left\{\begin{array}{cl}
n/2 & \mbox{ if \(x\equiv0\pmod{2}\),}\\
(3n-1)/2 & \mbox{ if \(x\equiv1\pmod{2}\).}
\end{array}
\right.
\]
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 m0=1 and mi = d+1 for 1 ≤ i < d and where we have the inequality
$$
m_0m_1\cdots m_{d-1}=(d+1)^{d-1}\lt d^d.
$$
So it seems certain that the sequence of iterates \(n, T_d(n), T_d^2(n),\ldots\)
always eventually enters a cycle and that there are only finitely many such cycles.
Clearly Td(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 kth and 2kth iterates for find a cycle element. We list the cycle with smallest absolute value as starting point.
Last modified 8th December 2020
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