Re: Collatz conjecture question

On 03/23/2025 03:40 PM, Ross Finlayson wrote:

On 03/23/2025 01:34 PM, vallor wrote:

On Sun, 23 Mar 2025 11:19:03 +0100, efji <efji@efi.efji> wrote in
<vron6n$23ve9$1@dont-email.me>:
>

Le 23/03/2025 à 03:53, vallor a écrit :

The Collatz conjecture has come up in comp.lang.c, and it got me
thinking
about it.
>
First, I'm not a mathematician, nor do I play one on TV.  But I wanted
to find out if there were any papers or other references that
have discussed the following:
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To compute the next number in a series
Odd numbers: N = 3N+1
Even numbers: N = N/2
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So it seems that for odd numbers, the next number in the series
will always be even; but for even numbers, the next number might
be odd or even.
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And that's what I'm wondering about:  has anyone ever explored
whether or not the even operation would tend to "dominate" a
series, and that is why it eventually arrives at 1?

>
Nobody knows (yet) if it always arrives at 1...
The strongest result on the subject is due to Terence Tao
https://arxiv.org/abs/1909.03562
and it is quite away from the proof of the conjecture.
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Numerically, a repartition of roughly 1/3 of odd numbers and 2/3 of even
numbers is observed, with a larger proportion of even numbers near
convergence. No proof at all for all this.
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Good luck :)

>
Thank you for the reply, very much appreciated.
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I also found this article:
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https://www.researchgate.net/publication/361163961_Analyzing_the_Collatz_Conjecture_Using_the_Mathematical_Complete_Induction_Method
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"Analyzing the Collatz Conjecture Using the Mathematical
Complete Induction Method"
>

>
There are models of integers with and without Szmeredi's theorem,
it's _independent_ usual laws of small numbers since there are
multiple models of integers, and of course a neat, simple, direct
logical argument that there's no standard model of integers,
only fragments and extensions.
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So, seeing this kind of conjecture decided one way or the other,
rather involves the modularity and infinitude of integers,
and other systems of numbers.
>
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"Complete Induction" then can be completed either way,
as if regards to, here, for example "not.first.false"
vis-a-vis "not.ultimately.untrue", like yin-yang ad infinitum.
>
>

It's like,
Q: Why did Cohen demonstrate Cantor's CH independent ZF?
A: Why not demonstrate it's inconsistent either way?
Q: What axiom should be added to ZF to decide CH some way?
A: What axiom should be removed?
The Russell-ian retro-thesis that there's an ordinary complete
inductive set, some accept with comfort, others distaste,
others don't.
Mirimanoff, by whom was coined "extra-ordinary", finds a
kindred spirit of sorts in Cohen, who uses it in, "forcing".
It's not ZF any more, ..., also not ordinary and
those aren't cardinals.
Number theorists variously do or don't posit a,
"point at infinity", and variously, a composite
or prime, or other features that would accompany
various result about the modular and regular and
dispersive and attenuative, and "super-tasks".
The regular singular points of the hyper-geometric
are zero, one, and infinity.
Sort of a, "do the math", as it were.