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 $L(n) = L(n+2)$.