Contents
Next: A heuristic argument.
Up: No Title
Previous: Introduction
The known results on the problem are most elegantly expressed in
terms of iterations of the function
One way to think of the problem involves a directed graph whose
vertices are the positive integers and that has directed edges from n
to .
I call this graph the Collatz graph of in honor of
L. Collatz [25].
A portion of the Collatz graph of is pictured in Fig. 1.
A directed graph is said to be weakly connected
if it is connected when viewed as an undirected graph,
i.e., for any two vertices there is a path of edges joining them,
ignoring the directions on the edges.
The Conjecture can be formulated in terms of the
Collatz graph as follows.
3x+1 CONJECTURE (First form).
The Collatz graph of on the positive integers is weakly connected.
We call the sequence of iterates
the trajectory of n.
There are three possible behaviors for such trajectories when n > 0.
- (i).
Convergent trajectory.
Some .
- (ii).
Non-trivial cyclic trajectory.
The sequence eventually becomes periodic and
for any .
- (iii).
Divergent trajectory.
.
The Conjecture asserts that all trajectories of positive n are
convergent.
It is certainly true for n > 1 that
cannot occur without some occurring.
Call the least
positive k for which the
stopping time
of n, and set if no k occurs
with .
Also call the least positive k for which
the total stopping time
of n, and set
if no such k occurs.
We may restate the Conjecture in terms of the stopping time
as follows.
3x+1 CONJECTURE (Second form).
Every integer has a finite stopping time.
The appeal of the problem lies in the irregular behavior of the
successive iterates .
One can measure this behavior using the stopping time, the total stopping
time, and the
expansion factor
defined by
if n has a bounded trajectory and if n has a divergent
trajectory.
For example n = 27 requires 70 iterations to arrive at the value 1 and
Table 1 illustrates the concepts defined so far by giving data on the
iterates for selected values of n.
TABLE 1. Behavior of iterates .
The Conjecture has been numerically checked for a large range of
values of n.
It is an interesting problem to find efficient algorithms to test the
conjecture on a computer.
The current record for verifying the Conjecture seems to be held
by Nabuo Yoneda at the University of Tokyo,
who has reportedly checked it for all
[2].
In several places the statement appears that A. S. Fraenkel has checked that
all have a finite total stopping time;
this statement is erroneous [32].
Contents
Next: A heuristic argument.
Up: No Title
Previous: Introduction