TITLE: What are some reasonable things to prove about the Collatz Conjecture?
QUESTION [9 upvotes]: I am writing an undergraduate paper on the $3n+1$ problem, and I am looking for some theorems related to the problem that would be reasonable for someone with my mathematical background to prove. I'm near the end of my undergraduate studies. I have had a pretty basic introduction to Number Theory and Group Theory. I have also taken a few introductory Computer Science courses. Those are the only courses that I've taken that seem to be relevant to the problem but otherwise my math classes have been: the usually calculus sequence, real analysis, introductory topology, linear algebra. 
A good answer to my question would do the following:

Either state one or more non-trivial theorems related to the Collatz conjecture, or reference sources that would contain this kind of information.
Would NOT state any proofs unless a proof or partial proof would be essential for someone with my requisites if they are to be able to provide a full proof.

Theorems that I have already proved:
If $L(n)$ is a function that counts the number of mappings in the Collatz function for $n$ to get to 1, then if $n = 8k+4$, $L(n) = L(n+1)$.
If $n = 128k+28$ then $L(n) = L(n+2)$.

REPLY [2 votes]: One important relation to other number theoretic problems is that of the distance of perfect powers of 2 and 3, for which a lot of material exists, even online accessible. The existence of the so-called "1-cycles"1 (which is a whole class of possible cycles of arbitrary length but a certain simple structure) is directly dependent on the question, whether $$ {k3^N-1 \over k2^N-1}=2^A $$ exists in natural numbers N,A and k (besides for "trivial" solution (N,A,k)=(1,1,1), which leads to the (unsolved) problem observed in the Waring-conjecture whether $$ (3/4)^N \lt \{ (  3/2)^N \} \lt 1-(3/4)^N $$ $ \qquad \qquad $ where $ \{ \cdot \}$ means the fractional part (see for instance mathworld)
Here is also the concept of K.Mahler's "z-numbers" relevant, which also relates the "1-cycles" in a general view to the powers of 3/2 and is also not too difficult to understand for a treatize which focuses an undergraduate publicum.
1Simons, John; de Weger, Benne, Theoretical and computational bounds for (m)-cycles of the (3n+1)-problem, Acta Arith. 117, No. 1, 51-70 (2005). eudml, ZBL1136.11019.