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.