Background: Here's a crash course on the Collatz Conjecture. Basically, you take a number and if it is even you divide it by two. If a number is odd, you multiply it by three and then add one. You keep doing this until the number becomes 1. The conjecture says that performing this operation for any number eventually results in 1. A quick note on the numbers used, they must be positive non-zero integers. As pointed out by Gottfried, there are contradictions otherwise. Stating the above in mathematical form,
$$n_{t+1} = \begin{cases} n_t/2, & \text{if $n_t$ is even} \\ 3 \cdot n_t+1, & \text{if $n_t$ is odd} \end{cases}$$
What I found: Here's what was kind of surprising to me. If you perform $3 \cdot x+1$, where $x$ is odd, then you will always get an even number. However, ${x \over 2}$ doesn't always result in an odd number, if $x$ is even. My assumption is this, if one could figure out how many times you have to divide an even number by two to get an odd number, headway could be made in proving the conjecture. However, when I went to find $v_2(e)$, where $e$ is an even number, and $v_2(e)$ is the number of times you can divide the number by 2, I discovered a fractal like pattern! If you plot the values of this function, you get,
$$1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,\ldots$$
The first value is $v_2(2)$. The second is $v_2(4)$, and so on. The sequence corresponds to sequence A001511.
If you define an L-system with,
$$C_n \rightarrow C_n (+)^{n-1} P (-)^{n-1} C_n$$
Where $P$ is a plot move right command, $+$ is a move up command, $-$ is a move down command, and $C_n$ is the string. The up/down commands just add/substract from a counter. The plot command will actually plot the value, and move the independent variable up one. The values of $v_2(e)$ can be extracted from $C_n$ by "reading" $C_n$ left to right and adding $1$. The exponents, denote moving up/down $n-1$ times. Graphically, we have,
Questions: Since this function $v_2(e)$ has a self-similar (fractal-like) nature, does that mean that the Collatz Problem is related to fractals? Could this be used to study the problem? Since fractals are inherently complex to compute in a linear fashion, could this explain why the conjecture is so hard to prove? I believe it explains some of the complex behavior of the iterates under the Collatz system.
Addendum:
Given an odd number, the next odd number in the Collatz iteration is given by,
$$(1) \quad o_{t+1}={{3 \cdot o_t+1} \over {2^{v_2(3 \cdot o_t+1)}}}$$
Since any even number fed into the iteration is reduced to an odd number under the iteration, we need only consider odd numbers. Therefore, the Collatz conjecture is equivalent to the conjecture that the Dynamical System $(1)$ has one and only one fixed point and that this fixed point is $1$ and attractive. In addition, the basin of attraction for this fixed point is the entire set of positive odd numbers. Now we can finally see relevance of the self-similarity of $v_2(o_n)$. Its highly complex behavior results in unusually a complex behavior for the Dynamical System.