I was discussing the Collatz Conjecture with a friend of mine who's an engineer and a physics nut, and he made an interesting point.
So I'll just assume we all know about choosing a positive integer n and then applying 3 n+1 to an odd integer and n/2 to an even integer.
My friend's point is, there is no upper limit to positive integers, so the rules for dealing with an initial n and its subsequent values become an infinite cycle, until we hit the 4-2-1 pattern that signals the end. But we know that as soon as we hit a value of n which is a power of 2, we're on track to reach that 4-2-1 cycle, because we're going to be reducing the exponent in unitary increments each time the rule is applied.
So my friend said, if the process is infinite, then the probability of reaching a value of n which is a power of 2 becomes 1; that is, it's an absolute certainty. If we're talking about an infinite process, then anything that can happen will happen; it's only a matter of time. Thus if we're definitely going to have a value for n which is a power of 2, then we're going to be able to reduce any positive integer we start with to 1, and the conjecture is proven.
There must be something logically wrong with this, because there's no way he's the first person to think of it, but I can't see what it is. I'm inclined to point out that, by the same logic, there must also be an infinite cycle of values of n which never 'land' on a power of 2.
So, my question is: is this a proof of the Collatz Conjecture? And if not, why not?
DISCLAIMER: I do not believe for one second that this is a proof of the Collatz Conjecture.