Summary of Question: Where can I find a discussion of attempting to prove the Collatz Conjecture via substitution and abstract examination?

I've done a lot of reading on the problem, including Lagarias' summary and addendum thereto. I am not a professional mathematician, so I don't know the language for what I'm trying to find, and the great GOOG / all the links I've found on MathOverflow here are not helping. I tried asking for advice in an unfocused format, so this is my question after more pondering and reading the FAQ.

To explain my basic approach and why I'm asking: my initial glance at the Conjecture involved, "A process described by $(2^0 + 2^{-1})(x+ 1/3)$ and $2^{-1}x$ that magically stays whole? Of course that's not going to go on forever. What do you mean it hasn't been proven?"

My examinations eventually led me to substitute (2n+1) for inputs, which assumes (and possibly shows) that the inputs that would cause strange situations are odd. This seems to lead to finding increasingly complicated rational coefficients not subject to binary carrying after a few iterations. Eventually - as far as I can tell - the only 2n+1 input that will work is zero; this happens because valid inputs for n need to have an increasing number of factors that run into a Zeno's Paradox-like wall. (for the function to consist entirely of whole numbers)

Because I can't see this as flawed I would like someone to explain to me why it is, or point me in the right direction. As I said, I have read quite a bit and Googled around, but I am not a mathematical scholar, (I hate proving the already proven. A lot.) and I am probably using the wrong terminology.

My apologies if I offend with such an untutored question.

doescontinue forever; the conjecture says that it eventually does so in a particular loop. – Andreas Blass Jan 6 '12 at 18:18