(i) Like “in Collatz”, an odd number n is transformed to 3n+1.

(ii) For an even number n a random decision is made: with probability 1/2, n is divided by 2 (like in Collatz), but with the other 1/2, n is divided by 2 and 1 is subtracted from the outcome. Subsequent random decisions are made independently of each other.

The modification in (ii) makes sure that for each even n, the result is even again with probability 1/2.

4 leads either to 2 or to 1, both with probability 1/2.

The expected number of iterations until 1 is reached for the first time is c*log(n_0), where “magic” 4/3 occurs in c somehow. ]]>

I notice that complex cycles occur only where it’s possible for the common differences of odd input-output numbers to be zero. That requires what I’ll call a zero-point equilibrium. What is that? It’s simply the n in 3x+n. It means that for 3x+17 the zero point is 17 – and for 3x+1 it’s 1, of course. For 3x+17, common differences have inverse pairs: +14, −14 and −20, +20. For the 3x+17 loop for 27, the addition of common differences yields 0 (+368-368). Such inverse-pair symmetry can never exist for 3x+1.

]]>It seems to me that this is inherent in considering arbitrary colorings and that some sort of order parameter(s) are missing from the story. e.g. the number of irregular pairs plays this role in Malliaris and Shelah’s analysis of the regularity lemma for graphs.

]]>Hi everyone. Although there is a new post by Professor Tao, I wanted to close my approach to the conjecture with a basic scheme that summarizes my approach. The numbers of the form (8) share structure with the difference of nth power, which to simplify I call it TCR, Tao-Collatz Reminder.

The conjecture needs to answer two questions:

-1. Are there periodic orbits?

-2. are there infinite orbits?

We know that the conjecture is true for many values. All reach 1 and there are no periodic orbits (except the trivial).

This approach to the conjecture through the notable product difference of nth power could serve to illustrate the mechanism of why the orbits reach 1, and also how the mechanism of the periodic orbits works and why they cannot occur for the powers of 2 and 3 .

Once this question was resolved, one could then attack the existence or not of infinite orbits, or at least, do it independently.Thanks ]]>

The arguments in my paper are unable to “see” anything going on in subsets of that are sparser than about in cardinality, so in particular would not be able to interact with the Krasikov-Lagarias set with the current level of bounds. Perhaps in the future one could potentially imagine the error terms in the almost all results and the lower bounds on the preimage results improving to the extent that they can start working with each other, although this possibility looks somewhat remote. However, there is a link between KL type bounds and uniform distribution of the Syracuse random variables in my paper, in that a (currently unproven) equidistribution hypothesis on the former can lead to improvements on the latter (in particular, to replace the 0.84 exponent by any exponent less than 1); I may detail this in a separate blog post.

]]>As a follow up question, can your result can be combined with the Krasikov-Lagarias result to get better estimates on how many numbers reach 1? This is terribly naive, and so must be wrong, but if there is a set S of density whose Collatz iterations can be shown to reach one, and the "generic" Collatz map reaches values much smaller than N, it seems like it might be possible to get some lower bounds on how many orbits hit S (and thus also go to one). Is there a reason that S and orbits of the Collatz map do not collide in a naive way?

]]>First I considered that there is a different cycle than 4.2.1, this cycle must have a maximum and minimum odd number

When analyzing the maximum odd number, I discovered that it must have the form 36n-19 or 36n-7

Using the collatz function, I discovered that one of the following numbers within the cycle should have the form 27n-14 or 27n-5

When using the “inverse function”, I discovered that the previous odd number within the cycle must have the form 24n-13 or 24n-5

Since these numbers are within the cycle, the numbers of the form 27n-14 or 27n-5 must reach the numbers of the form 24n-13 or 24n-5 using only the collatz function (multiply 3, add 1 and divide by one power of 2 indefinitely) which is clearly impossible

I did all this analysis graphically, but here in the comments I can’t put the graphics ]]>