### Blanton C. Wiggin's generalization of the 3x+1 mapping

The following mapping \(T_d:\mathbb{Z}\to\mathbb{Z}\) was defined in a paper *Wondrous Numbers - Conjecture about the 3x + 1 family*, Blanton C. Wiggin, J. Recreational Math. **20**, No. 2 (1988), 52-56.
Let \(d\geq 2\). Then
\[
T_d(x)=\left\{\begin{array}{ll}
\frac{x}{d} &\mbox{if \(x\equiv 0 \pmod{d}\),}\\
\frac{(d+1)x-j}{d} &\mbox{if \(x\equiv j \pmod{d}, 1\leq j\leq d-2\),}\\
\frac{(d+1)x+1}{d} &\mbox{if \(x\equiv -1 \pmod{d}\).}
\end{array}
\right.
\]
For example, \(d=2\) gives the \(3x+1\) mapping:
\[
T_2(x)=\left\{\begin{array}{ll}
\frac{x}{2} &\mbox{if \(x\equiv 0 \pmod{2}\),}\\
\frac{3x+1}{2} &\mbox{if \(x\equiv 1 \pmod{2}\).}
\end{array}
\right.
\]
\(T_d\) is an example of a *relatively prime* mapping, in the language of Matthews and Watts, where \(m_0=1, m_j=d+1, 1\leq j\leq d-1\) and where we have the inequality
\[
\prod_{i=0}^{d-1}\frac{m_i}{d}=\frac{1}{d}\left(1+\frac{1}{d}\right)^{d-1}<1.
\]
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.

We have \(T_d(j)=j\) for \(-1\leq j\leq d-2, d>2\) and Wiggin conjectured that every trajectory starting at a positive integer will eventually reach some \(j\) in the range \(1\leq j\leq d-2\).

The conjectural picture for negative input is less clear. For example, with \(d=3, 6, 8, 9, 10\) it seems that all negative trajectories will end up at \(-1\). However, for \(d=4\), negative trajectories may end up in the cycle \(-9, -11, -14, -18, -23, -29, -36\) as well as \(-1\).

Here is a **table** of d for which there are exceptional cycles.

*Last modified 13th March 2023*

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