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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.

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    $\begingroup$ I agree with Gerry Myerson's answer. You might begin by thinking about the "magically stays whole" in your initial reaction to the problem. The choice between the two formulas is made, at each stage, to ensure that the result is an integer. Neither magic nor non-trivial mathematics is involved there. And therefore the process does continue forever; the conjecture says that it eventually does so in a particular loop. $\endgroup$
    – Andreas Blass
    Commented Jan 6, 2012 at 18:18
  • $\begingroup$ I see why it stays whole. The $2^-1$ term is offset by the addition of 1 to create a carry. That said, this carry does not exist / work after the second iteration when you substitute, because you are working with 3n+2. I am finding that it can only continue forever given this substitution because in the case of 2n+1 where n = 0, 0 has an infinite number of factors. There are no other whole integers that can claim this. $\endgroup$
    – Noobermath
    Commented Jan 6, 2012 at 18:32
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    $\begingroup$ You might like to read this: terrytao.wordpress.com/2011/08/25/… $\endgroup$
    – Neil Strickland
    Commented Jan 6, 2012 at 18:50

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I read your question as asking for some place in the literature where someone has tried your approach and has explained why it doesn't work. Youre unlikely to find such a thing, as when mathematicians find an approach that doesn't work, they generally don't write it up. I'm afraid the best anyone can do for you is encourage you to write up your method, being careful to define all your terms, to make sure that each step follows from what preceded with nothing left out, and with any luck somewhere along the way you'll see for yourself what the problem is.

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