Walter Carnielli's generalization of the 3x+1 mapping
The following mapping Td: ℤ → ℤ was defined in an email from Professor
Walter Carnielli to Keith Matthews (26th December 2010).
Let d ≥ 2. Then
\[
T_d(n)=\left\{\begin{array}{cl}
n/d& \mbox{ if \(n\equiv 0\pmod{d}\),}\\
((d+1)n + d - i)/d& \mbox{ if \(n\equiv i\pmod{d}, 1\le i\le d-1\).}
\end{array}
\right.
\]
For example, d = 2 gives the 3x+1 mapping:
\[
T_2(n)=\left\{\begin{array}{cl}
n/2& \mbox{ if \(n\equiv 0\pmod{2}\),}\\
(3n-1)/2& \mbox{ if \(n\equiv 1\pmod{2}\).}
\end{array}
\right.
\]
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 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 \(n, T_d(n), T_d^2(n),\cdots\)
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
(i) Td(n) = n for n = -1,...,-(d – 1);
(ii) 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, Td(x), Td2(x),... where |x| ≤
R ⁄ 2 = 60000 ⁄ 2 for cycling by Floyd's method of testing for equality of kth and 2kth 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 6th December 2020
Return to main page