(Follow this link to see the definition of the Collatz function and to experiment with Gaston Gonnet's Fast Maple Code for computing the stopping time function for the Collatz function!) 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.
We call the sequence of iterates the trajectory of n. There are three possible behaviors for such trajectories when n > 0.
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 byif 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 . In several places the statement appears that A. S. Fraenkel has checked that all have a finite total stopping time; this statement is erroneous .