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Through messing around with some *Collatz values and trajectories, I came across values that seemed to rapidly increase and then 'rather quickly' fall back to one, like an explosion. (They all take less than 150 steps to reach one).

Here are the numbers I found interesting:

511

peak value in trajectory: 39364

takes 61 steps to reach 1

1023

peak value in trajectory: 118096

takes 62 steps to reach 1

32767

peak value in trajectory: 28697812

takes 129 steps to reach 1

65535

peak value in trajectory: 86093440

takes 130 steps to reach 1

The trajectories of 511, 1023, 32767 and 65535 graphed.

Is this a well-known phenomenon? Why do the values seem to sharply fall back down to smaller values?

As a nice little bonus, the first and second pair's stopping times seem closely related for some reason...

*"Collatz" refers to the Collatz Conjecture. For a quick refresher:

If an integer $n$ is odd, then multiply by 3 and add 1. If $n$ is even, divide by 2. Repeat the process. The Collatz Conjecture states every positive starting number eventually reaches 1.

Griffon Theorist697
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    It's not a coincidence that all are of the form $2^n-1,$. – dxiv Jan 22 '18 at 05:57
  • Don't put part of your question in the title. A person should be able to answer your question having only read the body. – AJY Jan 22 '18 at 05:59
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    @AJY Is the body poorly worded? If so, I am happy to fix it. I'm sorry if the title bothers you. I prefer to put a paraphrased version of my question up there because I can get wordy and I don't want to confuse people. – Griffon Theorist697 Jan 22 '18 at 06:05
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    @GriffonTheorist697 the word "Collatz" never appears in the body. – AJY Jan 22 '18 at 06:06
  • @AJY I added some new edits. If you have any other concerns, please let me know. Thank you! – Griffon Theorist697 Jan 22 '18 at 06:15
  • I don't understand why this is not downvoted like any other Collatz related question... What is so special about this one? –  Jan 24 '18 at 01:49
  • @NaturalNumberGuy When asking this question, I figured I was not the only person who stumbled upon this pattern. I had trouble figuring out if this was something accessible online research papers on the Conjecture covered or not (I struggle reading through the literature due to lacking formal training). – Griffon Theorist697 Jan 24 '18 at 02:14
  • @NaturalNumberGuy From what I know, the Collatz questions that get down voted are "proofs" that supposedly solve the Collatz Conjecture. The reasons I have seen range from people not doing their research and stating they found something, the proof in question was extremely informal and or lacked in clarity, the author used Google Translate to translate their thoughts, the author fails to ask a question or request feedback, and the general act of claiming to solve the Conjecture insults the work of serious mathematicians over the last 80 years or so. Usually a mix of all of the above. – Griffon Theorist697 Jan 24 '18 at 02:20
  • @NaturalNumberGuy I am well aware that I have no business claiming I found anything new related to the Collatz Conjecture. However, I am obsessed and would like to learn from my own amateur thoughts and ideas as I continue my course of study. If I actually find anything ground-breaking for the Collatz Conjecture, it won't be anytime soon, and I would have a deep or respectable understanding of most or all the material published on the problem. – Griffon Theorist697 Jan 24 '18 at 02:27
  • I don't know which author you are referring to who supposedly have insulted other mathematicians work over the last 80 years. If your answer is valid, then i do not know what people are smoking.

    Me however, have never claimed to have any proof of the Collatz conjecture, I have asked for collaboration or thoughts on the issues I have raised in relations to the Collatz sequences, but most of my threads on the subject have gotten downvoted anyway. Im keeping an low profile when it comes to this conjecture here on math exchange because its a complicated topic.

     –  Jan 30 '18 at 22:03
  • I find myself almost in a similar position like yourself, as an amateur mathematician im trying to learn things for my better understanding, but I have for years been computer programmer and working on some theories that keeps me motivated. My suspicion is that maybe some simplifications can be done in the Galois $GF(2^x)$ field, or rather some simplistic binary particle theory where there are just two possible states on an integer lattice and one state overcome the other, like we see when the iterated function converges to 1. –  Jan 30 '18 at 22:04

1 Answers1

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The property is long well known1). Per formula with some $A$ and odd $k$ $$a_1 = 2^A k -1 \to a_j = 3^A k -1$$ (with some $j$ depending on your definition of what is "one step")
The key is here that the value $-1$ is a fixpoint in the negative integers for the Collatz-transform.
If moreover $k=1$ then numbers of the form $3^A - 1$ have the primefactor $2$ to some easily to determine exponent: $$ \{3^A-1 ,2 \} = 1+ [A:2] + \{A,2\}$$ where the curly braces mean "valuation" (here the exponent of the primefactor $2$) and $[A:2]$ is a "Iverson-bracket" giving $1$ if $2 | A$ and $0$ else.
After that, $a_j = 3^{16} - 1 $ is followed by many $x/2$ steps (here $6$ steps): $$ \{3^{16}-1,2\} = 1 + [16:2] + \{16,2\} = 1+1+4 = 6 \\ a_j = 3^{16}-1 \to a_{j+6}={ 3^{16}-1 \over 2^6 }$$

You might be interested to look at the partial orbits of numbers of the forms $2^A k -5$,$2^A k -7$, $2^A k+1$ with some restrictions on the exponents $A$ ...

1) For instance I've seen it in an 1978-article by R. Crandall. But that's only the oldest original article I've found online - surely that property is so basic that it was known even earlier.
Gottfried Helms
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