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>:
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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.
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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?

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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 :)

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