Re: 3n+5 pures and impures!

On Apr 21, 12:26 pm, Danny <fasttrac...@xxxxxxxxxxxxx> wrote:
3n+5 pures and impures.

Mensanator,

A disscusion in a previous post
"Was -- The Collatz discrete primes -->Pure patterns",
where any counter example integers would have to
be pure.
You stated that they (could) be impure.

I haven't seen any reason why there couldn't be.


I still maintain that any (integers) outside the Colatz
tree in a hypothetical counter example would have
to be pure wheather it be the smallest odd integer,
also pure, or any integers invoved in this counter
example.

Any of the numbers in the counter example tree? Think about
this for a minute. A counter example loop cycle must have
at least 275000 elements (proved by Lagarias - see Mathworld).
Some say, due to the current status of the sequential seed
search project, that such a loop cycle is more likely to have
over 4 million elements. Let's compromise and say the counter
example loop cycle has 1000000 odd integers.

EVERY number in the loop cycle will be either 1(mod3) or
2(mod3) which means a branch spawns from EVERY odd number:

branh0 branch1 branch2 branch999998 branch999999
| | | | |
-->odd...odd...odd <999995 odds skipped>...odd...odd-->
|_______________loop cycle_____________________________|

Because of the loop cycle, EVERY number on EACH of the 1000000
branches is an ancestor to EVERY number in the loop cycle. On
the trivial tree, impure sequence vectors have to be connected
directly above the number in question. This isn't necessary when
a loop cycle is involved. Not only does the branch that spawns
from N have to be void of impure sequence vectors, but every
one of the 999999 parallel branches must also be void.

Now, you're telling me that NONE of these 1000000 branches can
manage an impure sequence vector? If every branch were 0(mod3)
that could happen, but 0(mod3) branches can't form loop cycles.


I know that you can find larger integers in the
3n+1 tree that are impure (hailstones) without
doing a sequential seed search which is very
interesting but the whole idea of pures is a
sequential seed search where seed (n) determines
the state of the next position (n+1).

Nevertheless, as long as sequnce vectors/hailstone functions
are equivalent, it's legal to do the calculations with them.
This is done all the time in math, and is called a transformation.
Logarithms are a perfect example of a transformation. Addition
of the logarithms of a set of numbers is equivalent to the
multiplication of said numbers. La Place Transformations can be
used to solve differential equations. The fact that I can
calculate impurity by applying sequence vectors rather than
discover it by sequential seed searching does not invalidate
my results.


If you can find all (n+1) that are impure

Well, we can't. It was shown there is no closed form way of
determing the first solution of every impure sequence although
we can find every element of an impure sequence once we know the
first.

then
you will automatically find all (n+1) that are pure.
I don,t believe this is the case.

And you're right, the best we can do is use modulo 18 to find
the purity status of 16 of every 18 numbers.

So in sticking with the sequential seed search
and my belief that all (n+1) is either a (1) impure
or a ( 0 ) pure.

They have to be one or the other.


Experimenting with 3x+5 the sequental list
of integers involved with the tree terminating
in the repeated loop of 8,4,2,1,8,4,2,1..

I'm going to have to study what you've got here.

But first, a word on terminology. It's easy to understand
what "the tree" means when there is only one. For other
systems, we need a more rigorous definition.

Every 3n+C system has a tree with the loop cycle 4C, 2C, C.
This tree is refered to as "the trivial tree" because the
Crossover Point of the sequence vector (always [1,2]) has
a denominator of 1 and is thus, trivially an integer.

In sytems where Collatz is true, "the trivial tree" is the
ONLY tree, so proper terminology must be that "the tree"
means "the trivial tree" since it is the only tree common
to all 3n+C systems.

This is what is meant by structure-centric, the definition
is determined by the sequence vector. Value-centric
definitions are thus, incorrect. In other words, the branch
of the powers of 2 (value-centric) is meaningless in 3n+3
or 3n+5. It is ALWAYS the sequence vector that determines
what "the tree" is. It is just a coincidence that the powers
of 2 happen to be the trunk in 3n+1.


1,2,4,8,9,16,18,32,36,41,53,64,69,72,82,106,107,111,
128,138,141,143,144,163,164,169,189,212,214,217,
219,222,231,247,256,263,276,281,282,286,287,288...

So in 3n+5, these are all examples from the counter example
tree(s), "the tree" only contains multiples of 5.

The correct way to refer to the Collatz Conjecture is
"Every number in 3n+C reaches a [1,2] loop cycle rooted at C."
The original conjecture, being value-centric, is useless.


Where, as in 3n+1, 2^n is always impure.

I thought we were only talking about odd integers.

Now entering each integer as a sequental seed
and bypassing all integers not shown
(trees outside the 3n+5) and designating
each bypassed integer as pure.

At this point, I've got to have to figure out what you're saying
and correct your terminology.

I'll get back to you later.


(*) An integer on tree(s) outside the 3n+5 tree.
(------0------) all zeros ' inclusive.
Status of (n+1) either a ( 0 ) pure of ( 1 ) impure.

1,2(*)4(***)8,9(******)16(***************)32
1,1(0)1(--0-)1,0,(-----0----)-1-(------------0-------------),1

(***)36(****)41(***********)53(**********)64
(--0--)0,(---0---)0,(---------0----------)0,(--------0--------),1

(****)69(**)72(*********)82(***********************)106
(--0---),0,(-0-),0,(-------0-------),1,
(------------------0---------------------),1

107(***)111(****************)128(*********)138(**)141
,0,,(--0--),0,,(-------------0---------------),1, (-------0--------),
0,,(-0--),0

etc.
(The format is screwed up in google but observe in
Drexel sci-math where I believe it is not)

So my argument is---

How else could you possibly express the pures or
impures in the 3n+5 tree?

Good question.


Dan


.

Relevant Pages

  • Re: 3n+5 pures and impures!
    ... You stated that they be impure. ... to be pure wheather it be the smallest odd integer, ... Any of the numbers in the counter example tree? ... Crossover Point of the sequence vector has ...
    (sci.math)
  • Re: 3n+5 pures and impures!
    ... You stated that they be impure. ... to be pure wheather it be the smallest odd integer, ... Any of the numbers in the counter example tree? ... Crossover Point of the sequence vector ...
    (sci.math)
  • Re: 3n+5 pures and impures!
    ... You stated that they be impure. ... to be pure wheather it be the smallest odd integer, ... the trivial tree, impure sequence vectors have to be connected ... Crossover Point of the sequence vector ...
    (sci.math)
  • Re: 3n+5 pures and impures!
    ... You stated that they be impure. ... to be pure wheather it be the smallest odd integer, ... the trivial tree, impure sequence vectors have to be connected ... Crossover Point of the sequence vector ...
    (sci.math)
  • Re: The Collatz discrete primes!
    ... doubling outside the loop. ... longer on the counter example tree than on the regular tree. ... example being pure or impure but my thought is that the ... smallest odd integer in the counter example path would have ...
    (sci.math)
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