Murgu Table2To3 as Collatz Conjecture Unique Solution began in 2023 autumn and
reach first form around of Jun 2024. But also was coming with a lot of validations
in 2024, 2025 and 2026 - as wasn't understood - need more explanations.
Collatz Conjecture Murgu Method and
New Formulas Legacy related to Collatz Conjecture
was heavy to accept,then, will try one more time here.
This will be maybe out of its historic, but will try to explain Mathematical and
Logic for what in June 2024 Murgu Table2To3 Solved Collatz Conjecture.
Even if historic is not this right as will be here, real solution start via
Restructure of Collatz Conjecture Domain via an
Group Theory Method
by grouping Positive Integers into Grids of 6 Elements (Integers) and then
every positive Integer will be included as one of Formulas :
(1+6i) - (2+6i) - (3+6i) - (4+6i) - (5+6i) - (6+6i) -- G.0
Next step was a logical analytic of those grids related to Collatz Conjecture Procedures
, and first temptation was (3x+1) itself. This born small hopes for Murgu Inverse Method.
Murgu Inverse Method Legacy
for Collatz Conjecture is natural one as is (3x+1)
inverse use and natural revelation offered by analytic testes formulas became Odds to
Odds direct relation.
Then if in (3x+1) = ((2k)L), note x as Q will get
a primary Murgu Inverse Method Formula.
(((2k)L)-1) = 3 Q
Primary as will see long analytic will come with 2 Murgu Inverse Method Formulas .
Turning back to Group Theory Contribution will observe any useful things as distinction
between Evens and Odds became a small key for Solution , then as
each Odd number have Infinity Collatz Images in Evens via
((2k)L)
This is a Legacy for Excluding for work only Evens and easing logical analytic, but also
is revealing into Group Theory G.0 Formulas write as
(1+6i) - 2(1+3i) - (3+6i) - 2(2+3i) - (5+6i) - 2(3+3i) -- G.0.1
G.0.1 - 2(1+3i) - 2(2+3i) - 2(3+3i) send My for a loss time to analyze (2+3i),
(3+3i) in a Collatz Logic, but was a losing time related to Collatz Conjecture as
we have accurate Method and Formulas to solve Collatz Conjecture.
Evens exclusion Legacy for Collatz Conjecture Analytic is natural and included in
((2k)L)
Now our Analytic will resume to Group Theory Grids containing for only
(1+6i) - (3+6i) - (5+6i) -- G.0.0
G.0.0 mean also first step to Table2To3 as Collatz Conjecture New Coordinate System.
Next step is Logical Analytic of G.0.0 members related to Murgu Inverse Method and then
legacy Collatz Conjecture Solution.
Logical Analytic of G.0.0 members related to Murgu Inverse Method
Golden KEY of Collatz Conjecture Solution
it started with attempts of using Primary Murgu Inverse Method for each G.0.0 members
general Form and after work we get 3 important rules.
1. Murgu Lema for (1+6i) which brought
((2(2k+2)[1+6i]) - 1) = 3 Qi (C.E.-1)
A New Collatz Conjecture Formula which will see have propriety of function and will
return for.
Legacy of this Formula-Function related to Collatz Conjecture Solution was in all
materials demonstrated and is simple and can be demonstrated now even by a High
School Student Math Skilled.
2. Murgu Lema for (5+6i) which brought
((2(2k+1)[5+6j]) - 1) = 3 Qj (C.E.-2)
A New Collatz Conjecture Formula which will see have propriety of function and will
return for.
Legacy of this Formula-Function related to Collatz Conjecture Solution was in all
materials demonstrated and is simple and can be demonstrated now even by a High
School Student Math Skilled.
3. A base key for Collatz Conjecture Solution
Collatz Conjecture Logical Closures (3 + 6i)
Revealed By Murgu Inverse Method and contained hide in Collatz Procedures.
Demonstration is simple as :
(((2k)(3L))-1) ¬ 3 Q
I included it in a simple demonstrative formulas as (3L) include all Odds divided by 3.
This wear now Collatz Conjecture Solution Legacy AND brought 3 Important Collatz Conjecture Rules.
I. All (3+6i) or (3L) , all Odds Divided one or more times by 3 do not have UP a
Collatz Connection.
II. A Collatz Conjecture Pattern started from an (3 + 6i) never
will meet second (3 + 6k).
A Collatz Conjecture Pattern started from an (1 + 6i) or
from an (5 + 6i) never will meet an (3 + 6k).
Those 3 rules are important for Collatz Conjecture Solution.
Next Step will make a return to Group Theory Method as we get new Structural
Logical rules.
As those new rules legacy for Collatz Conjecture Solution been demonstrated became clear
a new structure related to Collatz Conjecture Solution and first is Table2To3 as
Collatz Conjecture new Logical Coordinate System.
For it , following new rules we renamed
(1+6i) - (3+6i) - (5+6i) -- G.0.0
with distinctive formal names.
LET1i=(1+6i) - LDNi(3+6i) - LET2i=(5+6i) -- G.0.0
Where LET - Logical Eternal Triad (nodes) and LDN - Logical Dead Node .
LET's (Logical Eternal Triads Nodes), AND LDN name been chosen not by hazard , but by considering
Theirs role in Collatz Conjecture New Structure brought by Group Theory Grouping in Grids .
Logical Dead Nodes have double role, Logical closures and Connecting Down only and only A LET nodes ,
important for Logical Analytic.
Logical Eternal Triads Nodes (LET's) have triple role
1. Assure for Up Connection (first one Q0) way back to Unity via Collatz Procedures.
2. Assure Eternal via infinity (not Infinity) Qi (i >0) Collatz Conjecture State.
3. Even if Down Connection is offered by a Collatz Procedure, ante rules assure this connection will
be only and only another LET.
Now we are able to model a NEW Collatz Conjecture Structure which Respect Collatz Conjecture Procedures.
This is absolute defined by Collatz Conjecture new Coordinate System Table2To3 . Table as we work with discrete
values even if New Murgu LET's Formulas are Functions also.
Table2To3 as Collatz Conjecture New Coordinate System.
By hazard as work was going on, we defined
Axle Ox (X axis) as axle of all Odds Integer, then all LDN's
and LET's.
Axle Oy (Z axis) as all LET"S included.
By here -- Collatz Conjecture Murgu Inverse Method will play its role via New Murgu Formulas .
Murgu Table2To3 or Collatz Table2To3 are 9, but Important and
Collatz Conjecture Murgu LET's 2 New Formulas
((2(2k+2)[1+6i]) - 1) = 3 Qi (C.E.-1)
((2(2l+1)[5+6j]) - 1) = 3 Qj (C.E.-2)
Those are Murgu Table2To3 Formulas or Collatz Table2TO3 Formulas
Theirs Legacy over Collatz Conjecture is absolute and DEMONSTRATED by 2024.
As to follow graphic LET1's and LET2's will note those as T... . and will can write
((2(2k+2)T1i) - 1) = 3 Qi (C.E.-1.1)
((2(2l+1)T2j) - 1) = 3 Qj (C.E.-2.1)
Those are Infinity and both Linear.
AND reveal
Infinity Murgu Table2To3
as simple validator for Collatz Conjecture Unique Solution Murgu Table2To3 . (C.E.-1) and
(C.E.-2) viewed analytic also as (C.E.-1.1) and (C.E.-2.1) are
Collatz Conjecture Unique Solution Murgu Table2To3
Infinity Murgu Table2To3 Validate Murgu Table2To3 as Collatz Conjecture Unique Solution as SHOW
Mathematical via k and l = 0 to Infinity that Cover as formulas all Collatz Conjecture Nodes.
Evens are covered by
(2lT1i) and (2lT2i)
- THEN -
as answer -
(2lQ1) and (2lQj) .
Murgu Table2To3 Formulas as Infinity Murgu Table2To3 Cover all Positive Integers.
Collatz Unicity
Collatz Unicity leaded a time to confusion Unique when mean
Infinity Murgu Table2To3 by theirs form show in fact every Qi and Qj for
i=j=0
(First T1i and T2j connections are Unique.)
and assure into Collatz Pattern the way down to Unity .
This Property may be understood easy if now imply Marker USA Murgu Arrow for k=l=0 .
And Then Infinity Marker USA Murgu Arrows for k=l - 0 To Infinity.
But here, first, need to demonstrate
Murgu Table2To3 or Collatz Table2To3 Formulas by theirs Structural Form
is a Collatz Conjecture Unique Solution for Mathematicians .
For it to start this logical analytic.
As Linear Functions via Slopes for (C.E.-1.1) are Infinity neither one of those will intersect.
As Linear Functions via Slopes for (C.E.-2.1) are Infinity neither one of those will intersect.
Marker USA Murgu Arrows show clear for each Murgu Table2To3 for which k=l, 0 to Infinity,
(C.E.-1) do not intersect (C.E.-2) if take each marker separately. But if take now all in
the same Graph (Infinity Murgu Table2To3), any (C.E.-1) Intersect (C.E.-2) - see The Slopes.
This can mean Collatz Unicity exclusion.
Then need a profound logical analytic for, and to do it for second form of (C.E.-1) and (C.E.-2) --
(C.E.-1.1) and (C.E.-2.1).
Intersection mean :
((2(2k+2)T1i) - 1) = ((2(2l+1)T2i) - 1)
then - (2(2k+2)T1i) = (2(2l+1)T2j)
then - (2(2k+2-2l-1)T1i) = T2j
then - (2(2(k-l)+1)T1i) = T2j
then (2(2v+1)T1i) = T2j
v negative or positive doesn't matter.
Because, as we know T1i , also T2j are Odds Integers
Then :
All (C.E.-1.1) intersections with (C.E.-2.1) are not in Collatz Conjecture Nodal Points.
Or never (C.E.-1.1) will Intersect (C.E.-2.1) into a Collatz Node or Integer.
Collatz Unicity became a Collatz Conjecture RULE.
2024 Murgu Table2To3 - Collatz Conjecture Unique Solution.
Murgu Table2To3 - Infinity Murgu Table2To3 - Infinity Marker USA Murgu Arrows
Not Only Solved Collatz Conjecture - But extracted for 3 new Math Jewel.
1. Small one - Table2To3 - applicability .
2. Collatz Conjecture Functional Divergence - victory may born a new Math Domain - Functional
Divergence Study-.
3. For a Time one of biggest Science Jewel
Murgu 1D Array Index Infinity_Per_6
Or Collatz 1D Array Index Infinity_Per_6.
Where Infinity_Per_6 is a Reality.
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