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.
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