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You might find interesting various probabilistic plausibility arguments for the $\rm\:3n+1$ conjecture. For an introduction see Lagarias's survey excerpted below, and for more sophisticated arguments see his paper How random are the 3x+1 function iterates?.

1. Introduction.$\;$ The $\rm\:3x + 1\:$ problem, also known as the Collatz problem, the Syracuse problem, Kakutani's problem, Hasse's algorithm, and Ulam's problem, concerns the behavior of the iterates of the function which takes odd integers $\rm\:n\:$ to $\rm\:3n + 1\:$ and even integers $\rm\:n\:$ to $\rm\:n/2\:$. The $\rm\: 3x + 1\:$ Conjecture asserts that, starting from any positive integer $\rm\:n\:$, repeated iteration of this function eventually produces the value $\rm\:1\:$.

The $\rm\:3x + 1\:$ Conjecture is simple to state and apparently intractably hard to solve. It shares these properties with other iteration problems, for example that of aliquot sequences (see Guy [36], Problem B6) and with celebrated Diophantine equations such as Fermat's last theorem. Paul Erdos commented concerning the intractability of the $\rm\:3x + 1\:$ problem: "Mathematics is not yet ready for such problems." Despite this doleful pronouncement, study of the $\rm\:3x + 1\:$ problem has not been without reward. It has interesting connections with the Diophantine approximation of $\log_2 3\:$ and the distribution (mod $\rm\:1\:$) of the sequence $\rm\:\{(3/2)^k : k = 1,2,\ldots\}\:$, with questions of ergodic theory on the $\rm\:2\:$-adic integers $\rm\:\mathbb Z_2\:$, and with computability theory--a generalization of the $\rm\:3x + 1\:$ problem has been shown to be a computationally unsolvable problem. In this paper I describe the history of the $\rm\:3x + 1\:$ problem and survey all the literature I am aware of about this problem and its generalizations.

2.1. A heuristic argument. $\;$ The following heuristic probabilistic argument supports the $\rm\:3x + 1\:$ Conjecture (see [28]). Pick an odd integer $\rm\:n_0\:$ at random and iterate the function $\rm\:T(n) = n/2\ \text{if n is even}\:$ $\rm\:\text{else}\ (3n+1)/2\:$ until another odd integer $\rm\:n_1\:$ occurs. Then $\rm\:1/2\:$ of the time $\rm\:n_1 = (3 n_o + 1)/2,\ 1/4\:$ of the time, $\rm\:n_1 = (3 n_o + 1)/4,\ 1/8\:$ of the time $\rm\:n_1 = (3 n_o + 1)/8\:$ and so on. If one supposes that the function $\rm\:T\:$ is sufficiently "mixing" that successive odd integers in the trajectory of $\rm\:n\:$ behave as though they were drawn at random (mod $\rm\:2^k\:$) from the set of odd integers (mod $\rm\:2^k\:$) for all $\rm\:k\:$, then the expected growth in size between two consecutive odd integers in such a trajectory is the multiplicative factor

$$\frac{3}{2}^{1/2}\frac{3}{4}^{1/2}\frac{3}{8}^{1/2}\cdots \ = \ \frac{3}{4} < 1\:.$$

Consequently this heuristic argument suggests that on average the iterates in a trajectory tend to shrink in size, so that divergent trajectories should not exist. Furthermore it suggests that the total stopping time $\rm\:\sigma_x(n)\:$ is (in some average sense) a constant multiple of $\rm\:\log n\:$.

From the viewpoint of this heuristic argument, the central difficulty of the $\rm\:3x + 1\:$ problem lies in understanding in detail the "mixing" properties of iterates of the function $\rm\:T(n)\ (mod\ 2^k)\:$ for all powers of $\rm\:2\:$. The function $\rm\:T(n)\:$ does indeed have some "mixing" properties given by Theorems B and K below; these are much weaker than what one needs to settle the $\rm\:3x + 1\:$ Conjecture.