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:

To compute the next number in a series
Odd numbers: N = 3N+1
Even numbers: N = N/2

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.

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?

I don't even know how to express that mathematically. Hopefully
this isn't too naive a question. TIA for any pointers one can
provide.

Thanks,

--
-v

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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.

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:

To compute the next number in a series
Odd numbers: N = 3N+1
Even numbers: N = N/2

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.

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.

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.

Good luck :)

--
F.J.

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On Sun, 23 Mar 2025 11:19:03 +0100, efji <[email protected]> wrote in <vron6n$23ve9$[email protected]>:

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:

To compute the next number in a series
Odd numbers: N = 3N+1
Even numbers: N = N/2

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.

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.

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.

Good luck :)

Thank you for the reply, very much appreciated.

I also found this article:

https://www.researchgate.net/publication/361163961_Analyzing_the_Collatz_Conjecture_Using_the_Mathematical_Complete_Induction_Method

"Analyzing the Collatz Conjecture Using the Mathematical
Complete Induction Method"

--
-v

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Le 23/03/2025 à 21:34, vallor a écrit :

On Sun, 23 Mar 2025 11:19:03 +0100, efji <[email protected]> wrote in <vron6n$23ve9$[email protected]>:

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:

To compute the next number in a series
Odd numbers: N = 3N+1
Even numbers: N = N/2

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.

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.

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.

Good luck :)

Thank you for the reply, very much appreciated.

I also found this article:

https://www.researchgate.net/publication/361163961_Analyzing_the_Collatz_Conjecture_Using_the_Mathematical_Complete_Induction_Method

"Analyzing the Collatz Conjecture Using the Mathematical
Complete Induction Method"

Wow, thanks for the link !
Obviously a false paper, written by non-mathematicians, and published in
a predatory journal, making big money in publishing anything.

Of course they have not proven the Collatz conjecture using an
hypothetic "Complete Induction Method" :)

It is a very strong conjecture, still unproven, and the link I have
given is a recent paper from Terence Tao who is maybe of one the best mathematicians ever (including Euler, Gauss, Poincaré etc.).

--
F.J.

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Am 23.03.2025 um 21:59 schrieb efji:

[...] Terence Tao who is maybe of one the best
mathematicians ever (including Euler, Gauss, Poincaré etc.).

Well, at least one of the greatest contemporary mathematicians.

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Le 23/03/2025 à 23:40, Ross Finlayson a écrit :

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.

In summary: we have Hachel with his dumb "complex numbers", now an
inventor of "multiple models of integers". Let's just find some genius
of "new real numbers" and we could form a team in the psychiatric
hospital :)

BTW, I just found the homepage of a unfortunate guy named "Ross
Finlayson". He his forced to have the following disclaimer: "I am not
the "Ross A. Finlayson" who posts prolifically to the "sci.math" and "sci.space.policy" newsgroups. We’re not related".

Imagine a real person named "Richard Hachel" :(

--
F.J.

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Le 24/03/2025 à 02:47, Ross Finlayson a écrit :

On 03/23/2025 05:01 PM, efji wrote:

Le 23/03/2025 à 23:40, Ross Finlayson a écrit :

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.

In summary: we have Hachel with his dumb "complex numbers", now an
inventor of "multiple models of integers". Let's just find some genius
of "new real numbers" and we could form a team in the psychiatric
hospital :)

BTW, I just found the homepage of a unfortunate guy named "Ross
Finlayson". He his forced to have the following disclaimer: "I am not
the "Ross A. Finlayson" who posts prolifically to the "sci.math" and
"sci.space.policy" newsgroups. We’re not related".

Imagine a real person named "Richard Hachel" :(

Yeah, it's been like that since about 20 years.
It's one of the oldest unchanged pages still on the Internet.

Wow, you sound just like "infinite foul toot J.G.".

Finlay Mor was killed in the 14'th century at
the Battle of Pinkie by a cannonball.

Somewhere, in Scot-land, there's a
Lone Highlander's Grave.

I don't know that we're related, ....

Dig a little deeper and start finding my
tens and tens of thousands of posts.

And a long, long line.

Hachel's just talking about iterating roots,
it's just a thing, whereas my talk about
complex numbers is about gaps in the analyticity
of the usual association of the Argand diagram,
and about how division is under-defined, and,
there are others, and about my original analysis
with the "identity dimension" the envelope of
the integral equations of d'Alembert, Clairaut,
and the linear fractional equation.

Which is very close to diffraction,
a fraction of differences.

Anyways, indeed it is so that there's reasoning
why Russell's retro-thesis, is, generously, an
unjustified stipulation, and, is, a bit more directly,
justified against.

Tens and tens of thousands of essays in mathematics,
logic, and physics, in the short essay form.

Also there are thousands and thousands of volumes
in my library, or, a ton of books.

So anyways, do you know of Mirimanoff and his role
in the influences of the development of ZF set theory?
Have you read Cohen's on the independence of CH?

Do you have a clue? (Without asking your phone, ....)

Have you ever said the word "metaphysics" or strung
together "point at infinity"?

Others have, ....

Two distinct pathologies. Hachel is struggling against himself about
basic and elementary notions. You are more a "name dropping guy",
chaining savant words in a random way. I suggest giving you the same
room in the psychiatric hospital. He could teach you how
a=b =/=> a^2=b^2, and you could teach him a lot of strange words whose definition is far above your head but make you feel that your are the
smartest guy on earth. Who will win between you two ?


--
F.J.

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Le 24/03/2025 à 09:38, efji a écrit :

Le 24/03/2025 à 02:47, Ross Finlayson a écrit :
He could teach you how
a=b =/=> a^2=b^2

a=25 b=25

a=b

5²=25

(-5)²=25

Then -5=5

R.H.

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Le 24/03/2025 à 15:47, Richard Hachel a écrit :

Le 24/03/2025 à 09:38, efji a écrit :

Le 24/03/2025 à 02:47, Ross Finlayson a écrit :
He could teach you how
a=b =/=> a^2=b^2

a=25 b=25

a=b

5²=25

(-5)²=25

Then -5=5

Are you proud of being so dumb in front of the whole world ?
:)


--
F.J.

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Le 24/03/2025 à 15:47, Richard Hachel a écrit :

Le 24/03/2025 à 09:38, efji a écrit :

Le 24/03/2025 à 02:47, Ross Finlayson a écrit :
He could teach you how
a=b =/=> a^2=b^2

a=25 b=25

a=b

5²=25

so a^2 = b^2 = 25, as expected from a=b

(-5)²=25

Then -5=5

-5 is neither a nor b.

What you've written does not confirm your absurdity (a=b =/=> a^2=b^2), it
only shows that from a^2 = b^2 your cannot conclude that a = b.

Are you really that silly or is this just your usual hypocrisy? Both?

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Am 24.03.2025 um 16:27 schrieb Python:

Le 24/03/2025 à 15:47, Richard Hachel a écrit :

 (-5)² = 25

 Then -5 = 5

Huh?!

Did this idiot even went to school?

Hint at RH: The function f: IR -> IR defined with f(x) = x^2 (for all x
e IR) does not have an inverse. Hence from a² (with any a e IR) you
can't "derive" /a/ by applying a "sqrt" inverse function (since there is
no such function).

On the other hand there's the function sqrt: {x e IR: x >= 0} --> {x e
IR: x >= 0} defined in a way such that sqrt(a^2) = a for all a e {x e
IR: x >= 0}.

Hence we may get sqrt(25) = 5, but certainly NOT sqrt((-5)²) = -5.
<facepalm>

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Le 24/03/2025 à 16:27, Python a écrit :

Then -5=5

-5 is neither a nor b.

What you've written does not confirm your absurdity (a=b =/=> a^2=b^2), it only
shows that from a^2 = b^2 your cannot conclude that a = b.

Are you really that silly or is this just your usual hypocrisy? Both?

If you were intelligent, you would understand that our discussion isn't
simply about a=b and therefore a²=b².

There must be something more "complex" to understand.

We continue: i²=-1.

At which point Hachel comes in: "Yes, that's good, but it's not enough. It
must be said that we are now going to take control of mathematics, your television, and your bank account to pay for the war in Ukraine."

So I take control of the world's mathematics, and I set for all x, then,
for imaginary numbers, i^x=-1.

The die is cast.

I don't allow dispute.

And I set i°=-1, i²=-1, (i²)²=-1 as long as I like.

I set the obvious (you have to follow the concepts to the end): i=-1.

You set f(x)=x².

So we have f(-1)=1.

But you set f(i)=i²=-1, which is clearly contradictory.

So there's a mistake. Where's the mistake?

We are certain that i=-1.

We are certain that f(-1)=1.

We are certain that i²=-1.

What happens when we say that f(i)=1 when theoretically x²=i²=-1?

Isn't the mistake in naming f instead of g?

Think about it.

R.H.

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On 3/24/2025 10:47 AM, Richard Hachel wrote:

Le 24/03/2025 à 09:38, efji a écrit :

He could teach you how
a=b =/=> a^2=b^2

a=25 b=25

a=b

5²=25

(-5)²=25

Then -5=5

https://en.wikipedia.org/wiki/Affirming_the_consequent
⎛
⎜ Affirming the consequent
⎜
⎜ In propositional logic, affirming the consequent
⎜ (also known as converse error, fallacy of the converse,
⎜ or confusion of necessity and sufficiency)
⎜ is a formal fallacy (or an invalid form of argument)
⎜ that is committed when,
⎜ in the context of an indicative conditional statement,
⎜ it is stated that because the consequent is true,
⎝ therefore the antecedent is true.

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Le 24/03/2025 à 16:43, efji a écrit :

Le 24/03/2025 à 15:47, Richard Hachel a écrit :

Le 24/03/2025 à 09:38, efji a écrit :

Le 24/03/2025 à 02:47, Ross Finlayson a écrit :
He could teach you how
a=b =/=> a^2=b^2

a=25 b=25

a=b

5²=25

(-5)²=25

Then -5=5

Are you proud of being so dumb in front of the whole world ?
:)

Si tu savais comme je me préoccupe peu des pensées du monde entier.

R.H.

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On 3/24/2025 11:58 AM, Richard Hachel wrote:

Le 24/03/2025 à 16:27, Python a écrit :

I don't allow dispute.

'Nuff said.

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Le 24/03/2025 à 17:00, Richard Hachel a écrit :

Le 24/03/2025 à 16:43, efji a écrit :

Le 24/03/2025 à 15:47, Richard Hachel a écrit :

Le 24/03/2025 à 09:38, efji a écrit :

Le 24/03/2025 à 02:47, Ross Finlayson a écrit :
He could teach you how
a=b =/=> a^2=b^2

a=25 b=25

a=b

5²=25

(-5)²=25

Then -5=5

Are you proud of being so dumb in front of the whole world ?
:)

Si tu savais comme je me préoccupe peu des pensées du monde entier.

R.H.

This is a part of *your* problem, in addition to be stupid you are
dementedly arrogant.

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Le 24/03/2025 à 16:58, Richard Hachel a écrit :

Le 24/03/2025 à 16:27, Python a écrit :

Then -5=5

-5 is neither a nor b.

What you've written does not confirm your absurdity (a=b =/=> a^2=b^2), it only
shows that from a^2 = b^2 your cannot conclude that a = b.

Are you really that silly or is this just your usual hypocrisy? Both?

If you were intelligent, you would understand that our discussion isn't simply
about a=b and therefore a²=b².

It is. There is nothing "intelligent" in denying that a=b => a^2 = b^2

There must be something more "complex" to understand.

We continue: i²=-1.

At which point Hachel comes in: "Yes, that's good, but it's not enough. It must
be said that we are now going to take control of mathematics, your television, and
your bank account to pay for the war in Ukraine."

So I take control of the world's mathematics, and I set for all x, then, for imaginary numbers, i^x=-1.

The die is cast.

I don't allow dispute.

And I set i°=-1, i²=-1, (i²)²=-1 as long as I like.

Your "postulate" can be disputed, like all postulate. As it is
inconsistent (it leads to contradiction) it has to be rejected. PERIOD.

I set the obvious (you have to follow the concepts to the end): i=-1.

You set f(x)=x².

So we have f(-1)=1.

But you set f(i)=i²=-1, which is clearly contradictory.

So there's a mistake. Where's the mistake?

On your side.

We are certain that i=-1.

We are certain that f(-1)=1.

We are certain that i²=-1.

What happens when we say that f(i)=1 when theoretically x²=i²=-1?

Isn't the mistake in naming f instead of g?

Think about it.

There is nothing to think about your bunch of nonsense.

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Am 24.03.2025 um 16:27 schrieb Python:

it only shows that from a^2 = b^2 your cannot conclude that a = b.

Indeed.

Are you really that silly or is this just your usual hypocrisy? Both?

Btw. Mückenheim recently stated

1 & n => n+1

(following John Gabriels lead, using terms instead of statements in
connection with logical connectives).

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On 3/24/2025 12:09 PM, Ross Finlayson wrote:

On 03/24/2025 08:59 AM, Jim Burns wrote:

https://en.wikipedia.org/wiki/Affirming_the_consequent
⎛
⎜ Affirming the consequent
⎜
⎜ In propositional logic, affirming the consequent
⎜ (also known as converse error, fallacy of the converse,
⎜ or confusion of necessity and sufficiency)
⎜ is a formal fallacy (or an invalid form of argument)
⎜ that is committed when,
⎜ in the context of an indicative conditional statement,
⎜ it is stated that because the consequent is true,
⎝ therefore the antecedent is true.

Doesn't that round-file material implication?

No.

Consider P Q P⇒Q

All possible circumstances:
 P Q P⇒Q
 T T  T
 f T  T
 T f  f
 f f  T

----
Material implication:
 P, P⇒Q true
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
Q inferred

All still.possible circumstances:
 P Q P⇒Q
 T T  T
 ̷f ̷T  ̷T
 ̷T ̷f  ̷f
 ̷f ̷f  ̷T

Q can only be true.
Infer Q.

----
Affirming the Consequent
 P⇒Q, Q true
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
P inferred

All still.possible circumstances:
 P Q P⇒Q
 T T  T
 f T  T
 ̷T ̷f  ̷f
 ̷f ̷f  ̷T

P might be true and might be false.
Don't infer P.

Or, you just pick when it's so?

The defining goal of
this entire project called "logic"
is to not.pick.
So, no.
But really, really, really "no".

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Am 24.03.2025 um 18:11 schrieb Python:

Le 24/03/2025 à 16:58, Richard Hachel a écrit :

I set i° = -1, i² = -1, (i²)² = -1

Your "postulate" can be disputed, like all postulate. As it is inconsistent (it leads to contradiction) it has to be rejected. PERIOD.

Indeed.

From Hachel's postulates we get: i² = -1 and hence (-1)² = (i²)² = -1.
But (-1)² = (-1)*(-1) = 1 (and 1 =/= -1) (following "standard" math).

Again, from his claim "i^x = -1" (for all "numbers" x), we might get i =
i^1 = -1 (since 1 certainly might be a "number"). Hence: i = -1. But
then i² = i * i = (-1)*(-1) = 1 (following "standard" math). While one
of Hachel's postulates is i² = -1. [...]

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Am 24.03.2025 um 18:12 schrieb Python:

Le 24/03/2025 à 17:00, Richard Hachel a écrit :

Le 24/03/2025 à 16:43, efji a écrit :

Are you proud of being so dumb in front of the whole world?

Si tu savais comme je me préoccupe peu des pensées du monde entier.

Did he mean: "Si tu savais comme je me préoccupe peu des pensées?"

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Am 24.03.2025 um 16:27 schrieb Python:

it only shows that from a^2 = b^2 your cannot conclude that a = b.

Indeed.

Are you really that silly or is this just your usual hypocrisy? Both?

Btw. Mückenheim recently stated

1 & n => n+1

(following John Gabriel's lead, using terms instead of statements in
connection with logical connectives).

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Am 24.03.2025 um 20:23 schrieb Moebius:

Am 24.03.2025 um 16:27 schrieb Python:

it only shows that from a^2 = b^2 your cannot conclude that a = b.

Indeed.

Are you really that silly or is this just your usual hypocrisy? Both?

Btw. Mückenheim recently stated

         1 & n => n+1

(following John Gabriel's lead, using terms instead of statements in connection with logical connectives).

His intention was to express "the principle of induction", btw.

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