http://www-personal.ksu.edu/~kconrow/" [Broken](adsbygoogle = window.adsbygoogle || []).push({});

I meanreallyread 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 3n+7 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 thatallthe disjoint graph pieces have LDA Headers.

So, collecting LDA Headers into a setmust necessarilyinclude

the counterexamples on the APT.

AllAPTs for all 3n+C systemsmusthave 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 doesnotprove the

graph components of the underlying Collatz graph are not

disjoint.

Does any of this sound right?

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# Has anyone read Ken Conrow's Collatz Conjecture website?

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