The Collatz Conjecture and a Question of its Generalization

[Guest]

Hello, I just had a quick question on generalizing the Collatz Conjecture. I have some logic concerning other iteratives that I would like to and need help disproving. It is quite simple. I think a lot of us misinterpret what is happening with this process. Values seem to be arbitrarily moving up and down, but in reality they are "searching" for a [tex]2^{n}[/tex]. I'd like to make a comparison to an individual looking for their keys in a room. How lit the room is, is the coefficient of n. As coefficients increase the room gets darker and it takes an exponentially longer amount of time to find the keys. Now there are limits to what the coefficients can be if we want them to exhibit similar behavior to 3n + 1. Which we believe is convergent. (Here is the math part) We shall now take a look at the properties of 3n +1. We see that there is an odd coefficient of n and an odd addend. This is purposely done in order to make n even. (i.e. if we take an odd times an odd we will get an odd number back and if we add an odd number to it we will always get an even number back). Now logically, we can deduce that two things will happen either we get a 2^{n} and we win and find the keys or we get an odd and have to keep looking. Well, that off will become an even and we will eventually start to traverse a list of evens. As we do this we are bound to come across a 2^n. It just becomes more difficult as the 2^n which will be valid is limited by the coefficient. So in reality the difficulty of finding the keys goes up in orders of magnitude of 2^x^n. So in the end I guess I am really just giving intuition behind why I Conjecture that all formulas that are restricted by the details above (i.e. n+1, 3n+1, 5n+1, 7n+3...odd*n+odd) converge to 1.