Collatz Conjecture Complete Solution
Collatz Conjecture Nodes Murgu Formulas
((2^(2k+2)[1+6i]) - 1) = 3 Q_i
((2^(2l+1)[5+6j]) - 1) = 3 Q_j
Solved Collatz Conjecture For Mathematician

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but as those were revealed by Collatz New Structural Separation, New Logical and New Methods
For it will need Heavy Part Explained
To understand It need understood important parts as
Collatz Conjecture Murgu New Structure, New Logic Structural aproach and Murgu New Method.

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Collatz Conjecture Murgu New Structure
is about separation of positive Integers in Grids of 6 Integers each.
Then , as known (for reading Murgu Inverse Method ), started by checking Grids Elements for any common proprieties. First was a simple observation
Every Even in Grids fall via Collatz Procedure division with 2 in an Odd
Then excluded temporarlly from work.
Second observed our now Odds Elements in Grids function of Position in have any common proprieties.
Real, this observation was an absolute result of -
Collatz Conjecture Murgu Inverse Method
which will be presented after will introduce Grids New Structure.

Those Odds in Grid been named and identified function of position in grid as:
1 + 6i --- LET_1i
5 + 6i --- LET_2i
3 + 6i --- LDN_i
Not need explanations as clear express positions on grid i, and to be easy will use a short forms :
L_1i = (1 + 6i) -- (MCF.01)
L_2i = (5 + 6i) -- (MCF.02)
D_i = (3 + 6i) -- (MCF.03)
also not need demonstration
D_i cover all LDN forms as (3^k), ((3^k)L_1i ) and ((3^k)L_2i ).

Collatz Conjecture Murgu Inverse Method

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started from Collatz Procedure for Odds
(3x+1)= 2^k Q
and applied to Grid's Odds Integers brought 4 revelation and brought also new Common proprieties for grid elements.
first need remark in a primary form Murgu Inverse Method Formula is
(2^k O -1) = 3 Q -- (MCF.04) -- important
REVELATIONS Values

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1. Murgu Inverse Method lead Direct as result ODD to Odd (avoid 2 divisions ).
2. Murgu Inverse Method reveal,
Logical Dead Nodes don't have UP Collatz Connections

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can be considered closures and bring an important Collatz Conjecture Property as rules.
(demonstrated in old 2024 material)
2.1. Every Logical Dead Node - via Collatz Procedure will Fall only and only into LET.
2.2. Every Collatz patterns starting from a Logical Dead Node will not pass another LDN.
2.3. Do not exist 2 LDN to fall into same LET.
3. First LET in Grid (L_1i) will reveal its multiple up Collatz Connections Nodes , Only and only Formula (demonstrated in old materials)
(2^(2k+2)(L_1i) - 1) = 3Q_i -- k =0 to ∞
as L_1i are identified AS ( 1 + 6i ) can write it as
(2^(2k+2)(1 + 6i) - 1) = 3Q_i -- (MCF.05)

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4. Last LET in Grid (L_2i) will reveal its multiple up Collatz Connections Nodes , Only and only Formula (demonstrated in old materials)
(2^(2k+1)(L_2j) - 1) = 3Q_j -- k =0 to ∞
as L_2i are identified AS (5 + 6i) can write it as
(2^(2k+1)(5 + 6i) - 1) = 3Q_i -- (MCF.06)

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(MCF.05) AND (MCF.06) Murgu Formulas may be considered

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Murgu - Collatz Conjecture Odds to Odds Formulas .

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(MCF.05) AND (MCF.06) in essence, Solved Collatz Conjecture via theirs form and as Infinity Double Linear Functions which assure Collatz Unicity.
For every k=0 to ∞ , those are 2 Linear Functions , with different slope, falling into Unity Like an Arrow having Unity as head.
as now know , those arrows are Infinity and neiter one wear the same slope.

Collatz Connections Unique (Unicity) is included in (MCF.05) AND (MCF.06) FORM -

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This can be understood only By Mathematicians maybe .
But mean:
Every LET can have Multiples Collatz Connection (infinity not Infinity ) but every one treated as functional connections between 2 values (points), have at start point the same value but at the end is different and never repeat.
Legality and authority of use of
Murgu - Collatz Conjecture Odds to Odds Formulas.

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over Collatz Conjecture is assured in first place by even Collatz Procedure for Odds.
But in old Material we demonstrate this right for everyone.
Repeat, for a Mathematician, implicit Collatz Murgu Formulas, (MCF.05) AND (MCF.06) Cover All Positive Integers Collatz Connections

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But need to demonstrate Collatz Conjecture Solution for everybody, and for it Collatz Murgu Inverse Method brought more Scientific tools. Those also are Validations Of Collatz Conjecture Unique Solution.

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VALIDATORS :

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First is Murgu Table2To3 or Collatz Table2To3
Murgu_Table2To3 - Collatz Table3To3, Click here
Murgu Table2To3 or Collatz Table2To3 are (MCF.05) AND (MCF.06) for k=l = 0
and show now (MCF.05) AND (MCF.06) COVER all via Double Linear Functions falling in Unity like an Arrow with head in Unity.
Known as Marker USA Murgu Arrow

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Murgu Table2To3 was inspired by important fact revealed by
Collatz Conjecture Murgu Inverse Method

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A Logical Dead Nodes Do not have UP Collatz Connections (Odd To Odd) , but only Down , And Down Coonection is only and only a LET.
That let us as Mathematic to deffine Graphic represntation for (MCF.05) AND (MCF.06) (k=0) on an deformed referential
- Oy only Logical Eternal Triads - Ox LET and LDN -
In Link above you can see Marker USA Murgu Arrow , with remark arrow head is assured by Collatz Procedure as it is a pure poetic metaphor.

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Second is more Strong :
Now is clear (MCF.05) AND (MCF.06) for k=l=0 TO ∞ brought Infinity Double Linear Functions , evryone falling into Unity Like Arrows.
Infinity Murgu Table2To3 or Infinity Collatz Table2To3

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Known Now as Marker USA Murgu Arrows
Collatz Connections Unicity is included into Functions form as implicit demonstration.
We also come with a small example for :
Murgu_Table2To3 - Collatz Table3To3
grade 1,2 and 3
Click here

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Rights over Collatz Conjecture - Infinity Collatz Table2To3 - (Murgu - Collatz Conjecture Odds to Odds Formulas) been demonstrated all along in old materials for all (1+6i) and (5+6i) Positive Integers Grids Nodes.

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Third is a Math Beauty :
Collatz Conjecture One-Dimensional Array

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(MCF.05) AND (MCF.06) as Infinity Arrows
Infinity Marker USA Murgu Arrows
Reveal All Collatz Nodes as an One-Dimensional Array named un 2024
Collatz Conjecture Pocket Map

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To show it we make 2 Javascript programs : one to generate The Array and second to reveal Instant , by using generated array (functions of how many grids array cover) to reveal in part or totally Collatz Pattern for an Odd Integers. Javascript was split in 2 programs to make instant answer for Collatz Patterns.
For Collatz Conjecture One-Dimensional Array, Click here
or
For Collatz Conjecture One-Dimensional Array, Click here
Now those Third make Collatz Conjecture Solution Understable for evry Skilled In Math.

Collatz Conjecture Solved

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Via Multiple Solutions :
Collatz Conjecture Murgu Formulas (MCF.05) AND (MCF.06) for skilled Mathematicians
Collatz Conjecture Murgu Table2To3 or Collatz Table2To3 for skilled Mathematicians
Collatz Conjecture Infinity Murgu Table2To3 or Infinity Collatz Table2To3 for strong skilled in Math
Collatz Conjecture Marker USA Murgu Arrows for all skilled in Math

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IMPORTANT is :
Collatz Conjecture Solved

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and brought any Math Beauty
New structural
New Logical Methods
Marker USA Murgu Arrow as possible New Science Toll
Collatz Conjecture AS One-Dimensional Array
A Mathematics Beauty FOR
Earth Science And Culture Treasure .

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Collatz Conjecture One-Dimensional Array

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have at base 2 Javascripts Programs - one Array Generator - Second arrays handler for innstant Collatz Patterns .
For sure br
Collatz Conjecture One-Dimensional Array

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will be into future sustained on a strong server by
Intel or Apple or Google or CISCO or Microsoft or Oracle
- even Collatz Conjecture is solved now -
But as a our
Computational Power Show.

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Murgu_Table2To3 1st, 2nd 3rd Grades parts to image Murgu Collatz Web Structure.
I keept part of Murgu Table2To3 First Grade as comparative image.