Recall the well-known $3x+1$ conjecture: if $T(n)=(3n+1)$/$2$ for $n$
odd and $T(n)=n$/$2$ for $n$ even, repeated application of $T$ to
any positive integer eventually leads to the cycle $$\{1\to2\to1\}\hbox{.}$$
We study a natural generalization of the function $T$, where instead of
$3n+1$ one takes $3n+d$, for $d$ equal to -1 or to an odd positive
integer not divisible by 3.
With this generalization new cyclic phenomena appear, side by side
with the general convergent dynamics typical of the $3x+1$ case.
Nonetheless, experiments suggest the following conjecture: For any odd
$d \ge -1$ not divisible by 3 there exists a finite set of positive integers
such that iteration of the $3x+d$ function eventually lands in this set.
¶ Along with a new boundedness result, we present here an improved
formalism, more clear-cut and better suited for future experimental
research.
@article{1048515662,
author = {Belaga, Edward G. and Mignotte, Maurice},
title = {Embedding the $3x+1$ conjecture in a $3x+d$ context},
journal = {Experiment. Math.},
volume = {7},
number = {4},
year = {1998},
pages = { 145-151},
language = {en},
url = {http://dml.mathdoc.fr/item/1048515662}
}
Belaga, Edward G.; Mignotte, Maurice. Embedding the $3x+1$ conjecture in a $3x+d$ context. Experiment. Math., Tome 7 (1998) no. 4, pp. 145-151. http://gdmltest.u-ga.fr/item/1048515662/