has anyone read Ken Conrow's Colletz Conjecture website?

[Mensanator]

http://www-personal.ksu.edu/~kconrow/

I mean really read it, to where they understand it, not just
to the point where their eyes glaze over?

I see his site cited quite frequently, but I don't know if
I've ever seen any critique. I admit, I was intimiitaded by
his site at first. But over the years, I'm become increasingly
skeptical, especially now (Jul 2009) that he thinks he's finally
worked out the proof. He's still asking for help to formalize
his proof into a publishable format and I can't help him there.
We have corresponded by e-mail frequently in the past and I've
helped him with little things over the years like fixing his
faulty state machines.

But I can't see the validity of this interger density thing.
And he seems reluctant to talk about my objections. I would like
to think my point devastates his theory so much he's speechless,
but maybe he thinks it's so trivially wrong he won't waste his
time (although with the help I've given in the past, you would
think he would have the courtesy to tell me what's wrong).

So I'm soliciting other opinions since Ken seems reluctant to
offer his.

I'll try to keep this as concise as possible, I hope I get it
right.

Let's start with the basis of Ken's proof, the Left Descent
Assemblies (LDA). Ken breaks up his Collatz graph into branches
that start with odd numbers 0 (mod 3) and ends them on the
first place an odd number is followed by 3 consecutive evens.
This odd number is called the LDA Header. After an LDA Header,
the sequence merges into other LDAs until they all eventually
merge to 1. Ken points out that all LDA Headers are 5 (mod 8)
and if all LDA Headers are gathered into a set and used as the
root of something he calls the Abstract Predecessor Tree (APT),
he can prove that this tree has an integer density of 1 which
he claims proves that every positive integer is on the Trivia
Collatz Graph, thus, proving it.

Now, I don't have any problem with this integre density thing.
If Collatz is true, of course the Collatz graph contains all the
integers.

But he's not using the Collatz graph, he's using the APT.

And this is the nub of my gist. I say the truth of Collatz does
not follow from integer density 1 of the APT even if it's true.

Suppose I use 3n+7 instead of 3n+1.

Now, we know 3n7 fails the conjecture, it's Collatz graph has
multiple disjoint pieces and any number not on the Trivial Graph
is a counterexample.

We know its LDA Headers are 3 (mod 8).

We know that ALL the disjoint graph pieces have LDA Headers.

So, collecting LDA Headers into a set MUST NECESSARILY include
the counterexamples on the APT.

ALL APTs for all 3n+C systems MUST have an integer density of 1,
there's no where else for an integer to be. It simply does not
follow that Collatz is true just because its APT has an integer
density of 1, since 3n+7 has and it isn't true.

I say Ken has abstracted away the very thing he's trying to
prove, that integer density of the APT does NOT prove the
graph components of the underlying Collatz graph are not
disjoint.

Doeas any of this sound right?

[alfansome]

Yes. I think your critique is correct. He is proving something that is
true but it is not the Collatz conjecture

Al

[Mensanator]

On Sep 9, 12:44 pm, alfansome <alfans...@yahoo.com> wrote:
> Yes. I think your critique is correct. He is proving something that is
> true but it is not the Collatz conjecture

Thanks for your time. If I get a few more comments, I'll mention them
to Ken.

Tally so far:

aggrees: 1
not agrres: 0
eyes glazed over: 1

> > Doeas any of this sound right?- Hide quoted text -
>
> - Show quoted text -

[jillbones]

Under "Paradoxes"

"On the one hand, there must be an equal number of odd and even
integers in the Collatz predecessor tree for the conjecture to be true
because there are clearly an equal number of odds and evens among the
integers. (There's a 1:1 mapping of each odd number with its
immediately larger even number.)"

This doesn't seem right. If the number of odds and evens were equal,
the sequence would increase indefinitely.

Forgive me, if you have already noted this discrepancy.

regards, Bill J

[jillbones]

On Sep 7, 9:37 pm, Mensanator <mensana...@aol.com> wrote:

What happens if there are more than three consecutive evens between
two odd numbers?

> After an LDA Header,
> the sequence merges into other LDAs until they all eventually
> merge to 1. Ken points out that all LDA Headers are 5 (mod 8)
> and if all LDA Headers are gathered into a set and used as the
> root of something he calls the Abstract Predecessor Tree (APT),
> he can prove that this tree has an integer density of 1 which
> he claims proves that every positive integer is on the Trivia
> Collatz Graph, thus, proving it.
>
> Now, I don't have any problem with this integre density thing.
> If Collatz is true, of course the Collatz graph contains all the
> integers.

"3*n" only if the starting seed is "3*n"!

PMI, but what is "the Collatz graph"?

[Mensanator]

On Sep 9, 6:09 pm, jillbones <b92...@yahoo.com> wrote:
> Under "Paradoxes"
>
> "On the one hand,  there must be an equal number of odd and even
> integers in the Collatz predecessor tree for the conjecture to be true
> because there are clearly an equal number of odds and evens among the
> integers.  (There's a 1:1 mapping of each odd number with its
> immediately larger even number.)"
>
> This doesn't seem right.  If the number of odds and evens were equal,
> the sequence would increase indefinitely.
>
> Forgive me, if you have already noted this discrepancy.

No, I haven't noted it. Because I don't see any problem.
Funny things happen when you drag infinity into the picture,
and I, for one, will try to avoid it.

For example, take the standard Collatz graph:
... ... ... ... ...
128 42 40__13 12
64__21 20 6
32 10______3
16______5
8
4
2
1 (loops back to 4)

In 3n+1 this is the Trivial (and only, as far as we know), graph
structure. The 3n+7 also has a Trivial structure (in addition
to others). The Trivial _structure_ of 3n+7 (or any 3n+C) is
identical
to that of 3n+1:

... ... ... ... ...
n o p___q r
j___k l m
g h_______i
e_______f
d
c
b
a (loops back to c)

The value of any node on the Trivial graph of 3n+7 is found
by simply multiplying the equivalent 3n+1 node by 7. Since node
f is 5 on the 3n+1 graph, f is 35 on the 3n+7 graph.

Assuming every node on the Trivial graph maps to a Natural Number,
then 3n+7 graph has the same number of nodes as the 3n+1 graph, yet
3n+7 only has numbers divisible by 7, all the other numbers are found
on different graphs.

Do you think 3n+7 has more numbers than 3n+1?

Spooky, eh?

Oh, by the way, I assume Collatz is true for 3n+1 and, using that
assumtion, I have a real neat algorithm that can find the index of
congruence class 0 of a given C for any arbitrary Sequence Vector.
Works beautifully. Someday, maybe I'll find a use for that. :-)

[Mensanator]

The part designated an LDA Sequence has come to an end. What continues
after that is a different LDA. Such as this:

3, 10, 5, 16, 8, 4, 2, 1
| |
| LDA Header
LDA Leaf

I don't recall what he calls the 16-1 section, as it has no header.

What Ken does is seperate out the individual LDA sequences, puts all
their headers into a set and satifies himself that every number leads
to an LDA header, all of which are convergent on this set of LDA
headers.

This only works if Ken makes the prior assunption that there are no
counterexamples in 3n+1. He has made the thing he's trying to prove
an axiom.

>
> > After an LDA Header,
> > the sequence merges into other LDAs until they all eventually
> > merge to 1. Ken points out that all LDA Headers are 5 (mod 8)
> > and if all LDA Headers are gathered into a set and used as the
> > root of something he calls the Abstract Predecessor Tree (APT),
> > he can prove that this tree has an integer density of 1 which
> > he claims proves that every positive integer is on the Trivia
> > Collatz Graph, thus, proving it.
>
> > Now, I don't have any problem with this integre density thing.
> > If Collatz is true, of course the Collatz graph contains all the
> > integers.
>
> "3*n" only if the starting seed is "3*n"!
>
> PMI, but what is "the Collatz graph"?

This thing, extended to infinity:

... ... ... ... ...
128 42 40__13 12
64__21 20 6
32 10______3
16______5
8
4
2
1 (loops back to 4)

According to Graph Theory, this a properly called a graph, not a
tree as "tree" is acyclic by definition. You can speak of Collatz
in graph theory as being true if it includes all positive integers
and that no nodes are disjoint.

3n+7 would, ideed, include all the integers, but the nodes would be
disjoint. For example, any number divisible by 7 could only connect
to other numbers divisible by 7, so the conjecture fails for this
reason.

>
>
>
>
>
> > But he's not using the Collatz graph, he's using the APT.
>
> > And this is the nub of my gist. I say the truth of Collatz does
> > not follow from integer density 1 of the APT even if it's true.
>
> > Suppose I use 3n+7 instead of 3n+1.
>
> > Now, we know 3n7 fails the conjecture, it's Collatz graph has
> > multiple disjoint pieces and any number not on the Trivial Graph
> > is a counterexample.
>
> > We know its LDA Headers are 3 (mod 8).
>
> > We know that ALL the disjoint graph pieces have LDA Headers.
>
> > So, collecting LDA Headers into a set MUST NECESSARILY include
> > the counterexamples on the APT.
>
> > ALL APTs for all 3n+C systems MUST have an integer density of 1,
> > there's no where else for an integer to be. It simply does not
> > follow that Collatz is true just because its APT has an integer
> > density of 1, since 3n+7 has and it isn't true.
>
> > I say Ken has abstracted away the very thing he's trying to
> > prove, that integer density of the APT does NOT prove the
> > graph components of the underlying Collatz graph are not
> > disjoint.
>

> > Doeas any of this sound right?- Hide quoted text -
>

> - Show quoted text -- Hide quoted text -

[jillbones]

On Sep 9, 5:57 pm, Mensanator <mensana...@aol.com> wrote:
> On Sep 9, 6:09 pm, jillbones <b92...@yahoo.com> wrote:
>
> > Under "Paradoxes"
>
> > "On the one hand,  there must be an equal number of odd and even
> > integers in the Collatz predecessor tree for the conjecture to be true
> > because there are clearly an equal number of odds and evens among the
> > integers.  (There's a 1:1 mapping of each odd number with its
> > immediately larger even number.)"
>
> > This doesn't seem right.  If the number of odds and evens were equal,
> > the sequence would increase indefinitely.
>
> > Forgive me, if you have already noted this discrepancy.
>
> No, I haven't noted it. Because I don't see any problem.
> Funny things happen when you drag infinity into the picture,
> and I, for one, will try to avoid it.

I meant Conroy's contention that there is always an equal number of
odds and evens. You were asking what people
thought of Conroy's paper and that seemed like something
that should not go unnoticed.

Perhaps I should have said, "will keep on getting larger"?

I don't understand the question. You have already shown
that they both have the same number of numbers. Somehow,
I think that this drags infinity into the picture!

> Spooky, eh?

It would be "spooky" if I thought that 3n+1 had the most
numbers, while it was 3n+7 that actually had the most.

> Oh, by the way, I assume Collatz is true for 3n+1 and, using that
> assumtion, I have a real neat algorithm that can find the index of
> congruence class 0 of a given C for any arbitrary Sequence Vector.
> Works beautifully. Someday, maybe I'll find a use for that. :-)

Probably long before I understand what you are talking
about

[jillbones]

Some 3n+7 sequences contain the loop "5 22 11 40 20 5".
Is this loop considered to be a trivial loop? Is it
possible to build a Collatz graph based on this loop?

[Mensanator]

Should be "5 22 11 40 20 10 5".

> Is this loop considered to be a trivial loop?

No, a Trivial Loop is defined by it's structure. Using 3n+C, that
structure is: (3n+C) -> (n/2) -> (n/2) -> (3n+C).

And the Trivial Loop will always be 4C -> 2C -> C -> 4C.

For C=1, you get 4 -> 2 -> 1 -> 4.

For C=7, you get 28 -> 14 -> 7 -> 28.

Any number not on The Trivial Loop is on a non-trivial loop.
Only multiples of C can be on the Trivial graph component whose
root is always 4C 2C C 4C. Keep in mind, I'm only talking about
when C is an odd number that is NOT a power of 3.

> Is it
> possible to build a Collatz graph based on this loop?

Sure. Start with your loop cycle

5_22
| 11__40
| 20
|_____10

Extend the branches vertically by multiplying by 2

... ...
176 640
88 320
44 160
5_22 80
| 11__40
| 20
|_____10

and attach branches where a node-7 is divisible by 3.

... ...
176 640_211
88__81 320
44 160__51
5_22 80
| 11______40
| 20
|_________10

Repeat extending branches and adding branches. Note there are many
numbers less than a hundred missing. They will be found on the
Trivial Loop (if a multiple of 7) or else on other graph components.

ALL the disjoint graph components taken together comprise 3n+7.
The Collatz Conjecture effectively states that ALL positive integers
are nodes of a single graph component. Obviously, this fails for 3n+7,
but (as far as we know) works for 3n+1, 3n+3, 3n+9, 3n+27, or any
where
C is a power of 3.

[Mensanator]

On Sep 10, 2:18 pm, jillbones <b92...@yahoo.com> wrote:
> On Sep 9, 5:57 pm, Mensanator <mensana...@aol.com> wrote:
>
>
>
>
>
> > On Sep 9, 6:09 pm, jillbones <b92...@yahoo.com> wrote:
>
> > > Under "Paradoxes"
>
> > > "On the one hand,  there must be an equal number of odd and even
> > > integers in the Collatz predecessor tree for the conjecture to be true
> > > because there are clearly an equal number of odds and evens among the
> > > integers.  (There's a 1:1 mapping of each odd number with its
> > > immediately larger even number.)"
>
> > > This doesn't seem right.  If the number of odds and evens were equal,
> > > the sequence would increase indefinitely.
>
> > > Forgive me, if you have already noted this discrepancy.
>
> > No, I haven't noted it. Because I don't see any problem.
> > Funny things happen when you drag infinity into the picture,
> > and I, for one, will try to avoid it.
>
> I meant Conroy's

Conrow's.

> contention that there is always an equal number of
> odds and evens. You were asking what people
> thought of Conroy's paper and that seemed like something
> that should not go unnoticed.

It's just that _here_ I was specifically asking for opinions
on LDAs and Ken's contention that it prooves Collatz. I'm not
saying there may not be other problems.

>
> Perhaps I should have said, "will keep on getting larger"?

I don't see the relationship to evens and odds being equal.
On the surface, it seems wrong, but maybe I'm not following
what you're talking about.

If you naively consider the pigeon hole principle and 3n+1 has the
same pigeon holes as 3n+7, yet 3n+1 has ALL natural numbers in its
pigeon holes but 3n+7 has only multiple of 7 in its pigeon holes,
doesn't it seem that 3n+7 must have more numbers from its extra
graph components?

> You have already shown
> that they both have the same number of numbers. Somehow,
> I think that this drags infinity into the picture!

Absoultely.

>
> > Spooky, eh?
>
> It would be "spooky" if I thought that 3n+1 had the most
> numbers, while it was 3n+7 that actually had the most.

Infinity always catches up in the end. I think concern over evens
and odds won't matter.

>
> > Oh, by the way, I assume Collatz is true for 3n+1 and, using that
> > assumtion, I have a real neat algorithm that can find the index of
> > congruence class 0 of a given C for any arbitrary Sequence Vector.
> > Works beautifully. Someday, maybe I'll find a use for that. :-)
>
> Probably long before I understand what you are talking
> about

That was intended to be obscure. :-)

> > > - Show quoted text -- Hide quoted text -
>
> - Show quoted text -- Hide quoted text -
>
> - Show quoted text -- Hide quoted text -