Is this a possible part of the solution to the Collatz Conjecture

Question

I'm not trying to prove the Collatz Conjecture, but I think I may have found a pattern that could help solve it. This looks at the number of times a number will go up and down before reaching 1. For example, 2 goes up no times and down once to reach 1. This could be notated as 2 = 0 # 1. Now, look at 5. 5 goes up once and down 4 times. This could similarly be notated as 5 = 1 # 4, which also equals 1 # (0 # 2). Clearly, 5 goes on longer than 2 in the Collatz sequence. 5 also goes on longer than 2 in this sequence. Now, think of a number, n, which equals n # n. This would also make this number equal to (n # n) # (n # n). This number would go on infinitely in a Collatz sequence and in this sequence of up and downs. n, though, wouldn't be a whole number because if a number went to infinity or stayed in a loop, it would go up and down infinite times. There are several other cases in this with two or three variables, but I was just curious if this idea even worked.

Answer 1

There's precisely one number among all the 2-adic numbers having the representation $n\textrm {hash} n$ and it's $-1$.

This follows from the fact the $(3x+1)/2$ conjecture is topologically conjugate to the tent map, which in turn means there are precisely two fixed points, and these are easily shown to be $-1$ and $0$.

Zero doesn't translate into $n\textrm {hash} n$ when going $3x+1$ but $-1$ does.

The two points of period $2$ in the $(3x+1)/2$ conjecture are period $3$ in the $3x+1$ conjecture so can't be $n\textrm {hash} n$ either.

What you're writing about is the relative frequencies of downs vs ups and this has been analysed at length. You can take it a little further with Littlewood Offord theory, which Terrence Tao does in his 2007 blog on the subject.

Some of the greatest real analysis experts in the world have worked on this and the concensus seems to be that analysis just doesn't have the tools to resolve down to fine enough resolution to prove the conjecture. One way of looking at this, is that the relative frequencies of the ups and downs don't quite hold enough information to show where a sequence goes, and that you need to somehow use the full information of the the actual sequence and order of ups and downs in order to resolve where it goes.

This is typical of some of these longstanding problems, they are often sharp in the sense that you cannot throw information away along the way, all the info needs to be used.

Answer 2

If, for some positive integer $n$, $n$ matches $n\ \#\ n$, then, by definition, the Collatz trajectory of $n$ reaches $1$ in $2n$ steps (i.e. in a finite number of steps), and $n$ is not a counterexample to the Collatz conjecture.