Explorations of the Collatz Conjecture - Moravian College

Chapter 2
Total Stopping Time
Recall that the total stopping time σ ∞ (x) of a positive integer x is defined to be the
number of iterations needed for that integer to reach one. In order to get a better
visual representation of total stopping time, we can construct a tree to represent
the function. Since we conjecture that all positive integers have trajectories that
contain 1, the integer 1 will be the root of our tree. In order to generate the tree
by starting with 1, we must use the inverse map of our function. We define the
inverse map as shown below.
T −1 (x) =
{
2x,
}
2x − 1
3
We will call T −1
0
−1 2x−1
(x) = 2x the even piece of our inverse map and T (x) =
the odd piece. On the tree, the even inverse is represented by vertical connections
between positive integers. In other words, the integer T −1
0
(x) is drawn directly
above the positive integer x. We will refer to these vertical connections as trunks
of the tree. New branches coming off these trunks are found using the odd piece
of the inverse map. A branch only occurs when T −1
1
(x) is equal to a positive
1
3
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