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Collatz Conjecture Solved
AS Beauty Logical
AND Murgu Conjecture Vicious Redundancy
Maybe never solvabel AND Eternal Mathematics Provocation
Via Mathematics Beauty - Math Leaf Effects -

MURGU CONJECTURE VICIOUS REDUNDANCY
CREATED, With Murgu_Table2To3 AND Marker_USA_Murgu_Arrow in Initial Form (in Collatz Coonjecture usage) or addapted to Murgu_CVR
Murgu Math Beauty Trio

MURGU CONJECTURE VICIOUS REDUNDANCY
Was inspired by Collatz Conjecture as mirroring it in Negatives and passed in pozitive as (3x-1).
This idea helped Collatz Conjecture Solution AND
lead to a beauty discovery Multiples , maybe to infinity (not Infinity) Roots via
Math Leaf Effect.

Instead of Collatz Conjecture , Murgu Conjecture Vicious Redundancy comme with 10 or more Roots and 2 or more Leaf Effects.
For Murgu Conjecture Vicious Redundancy can be adpted with small modifications Collatz Conjecture Murgu Formulas, Collatz Conjecture Murgu New Concepts, Collatz Conjecture Murgu , Logic and even can make for analyze a Murgu Table2To3 but for Murgu Conjecture Vicious Redundancy.
If Need to know about those ,

Here can see it
Here Can See It .

We Know The Motivation
Math Poetic Metaphor Murgu Arrow Loss
In fact is loss Arrowhead.
The Math Problem is HOW MANY ROOTS and HOW MANY LEAF EFFECTS.
We determinated 10 roots and 2 leaf effects.

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Repeat, Collatz Conjecture is Solved and Solution DEMONSTRATED at every level at
Collatz Conjecture End
To understand and to see everything demonstrated mayb need to luck around more times - BUT IS THERE -

All Murgu new concepts are needed , also Murgu Formulas , but over all
Inverse Logic
which have as base Murgu Inverse Method AND as result
Collatz Unique Solution
Murgu_Table2To3
guarded by
Marker_USA_Murgu_Arrow
wich show Murgu_Table2To3 small asymmetry as an ARROWHEAD in Unity.
Instead, Murgu Conjecture Vicious Redundancy do not posses this Arrowhead which to close Murgu Conjecture Vicious Redundancy Murgu_Table2To3 in an arrowhead to Unity.
Murgu Conjecture Vicious Redundancy HELPED In pure sense The Solution for Collatz Conjecture but also
BY ADDAPTING All Collatz Conjecure Murgu Concepts and Formulas
LEAD TO

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Murgu Math Beauty Trio

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Murgu Conjecture Vicious Redundancy , Murgu_Table2To3 and Marker_USA_Murgu_Arrow addapted for both reveal an
Mathematics Beauty
and an Earth Galactic Science Provocation. At this time Murgu Conjecture Vicious Redundancy don't have a solution, but maybe a Human Mathematician will get an answer. Until Then
Murgu Conjecture Vicious Redundancy
Earth Galactic Science Provocation.

This material keept yet as working material and rights to be revised for ORGANIZED and Brought to easy understanding form.
Down is old material about Murgu Conjecture Vicious Redundancy (Murgu_CVR), and a promise, as soon time will permit, Collatz Conjecture and Murgu Conjecture Vicious Redundancy will be revealed in ordonated forms!
IF WISH TO UNDERSTAND BEUTY OF
Murgu Conjecture Vicious Redundancy
pay attention to
Logical Inversion
may be util in more cases!

Three as Prime Number, Numbers Theory Property.

In, Murgu Conjectures Vicious Redundancy, Collatz Conjecture and Numers Theory is a , to say , lema

For every L pozitive Integer not diect connected to (i∗3):
odd (L=2K+1) we have
1a. (2^L + 1 = 3 x)
and for every L even (L=2K) we have
2a. ((2^L - 1 = 3 y))
, which seem to be a logical property of Number 3, maybe important for Numbers Theory but also for Prime Numbers logic curiosity, but a explanation for Collatz Conjecture and Murgu Conjecture Vicious Redundancy.
Where x and y are also pozitive Integers.

I said then recurence is enough , but to tray to demonstrte it using Recurrence .
L pozitive Integer odd (L=2K+1) we have
1a. (2^L + 1 = 3 x)
If (2^{(2K +1)} + 1 = 3 x)
then also must to be true
(2^{(2K +3)} + 1 = 3 z)
writing (2^{(2K +3)} + 1 = 3 z) as (2²2^{(2K +1)} + 4 -3 = 3 z)
and then as (2²(2^{(2K +1)} + 1) = 3 + 3 z)
that mean
2²(3 x)= 3(z+1)
Demonstration Complete because for (2a.) The Same.
This new property can to born in Numbers Theory a new Contest.
But also can born an appeal. Anyway I didn't saw this property so clear and evedinent as Collatz Conjecture and Murgu Conjectures Vicious Redundancy expose it .
My attraction for Collatz conjecture come from a bit "logic smell" of a Divergent Conjecture , but only a bit.
Also born a new problem :
All Pozitie Integers divisible with 3 pay respect for those lemas related how Collatz Conjecture Evolve starting from a k?
The answer is not and is contained into demonstration. Next pozitive Integer divisible with 3 will be (3x + 3), and next revealed pozitive Integer by lemas will be one from second lema :
(2^(2K+2) - 1 = 3 y)
We know betwen 2^K Integers and 2^(K+1) Integers we have another
2^K Integers , THEN:
FOR EVERY K , WE HAVE NON RESPECT direct FOR LEMAS ANOTHERS :

2(K-1)

------------

3

Then we have now a new property :
(2(K-1)) / 3 as numbers of Integers divisible with 3 between lemas at a Collatz Step.
Murgu Lemas.
Sorry , if I returned second day with any explanations.
Up we demonstrated Infinity Positive Integers pay respect for Collatz Conjecture and Murgu Conjecture Vicious Redundancy . For last even wearing new aspects of new "roots" 5,17 .. , but can answer if far away will be anoter one or more.
When I spoke about NON RESPECT direct to Lemas , is about 3 divisibility folowing Collatz Conjecture, and to remark also 2 new properties :
For every Positive Integer wich can't be write as (i*3) we have:
1. For every odd Integers L - (2^L+1) is divided by 3 .
2. For every even Integers L - (2^L-1) is divided by 3 .
Murgu Lemas
that explain Infinity Numbers pay respect for Collatz Conjecture and Murgu Conjecture Vicious Redundancy
as Number 3 Properties into NUMBER THEORY.
Those 2 Lemas explain also Collatz Conjecture , but explanation to be complete, also Murgu Conjectures and down logical explanation Complete it.
Between 2 Consecutive numbers divisible with 3 do not exist another divisible with 3 , and it is contained also in Lemas.

Collatz Conjecture have its beuty

but isn't a Paradox.

Because I have a small defect of speach, all will be write, and Math do not need speach.
I supose everybody know about Collatz Conjecture :
For every Integer will born a Collatz tree with tree root into unity after will follow next operations . If Integer I is odd then will be multiplaied with 3 and at result added 1, (result will be a even Number) then will follow one or multiple simplify with 2 until result will be a odd number or 1 .If result will be different of 1 process will be repeated until 1 . For I even first will be division , followed if is the case by the same tretment as above.
A Mathematics Tree every time is a beauty, because branches and nodes (connections with trunk) are logical connexions and can reveal any rules and properties on.
But over all this is a Powers property , via a Method to bring every integer at a form (2ⁿ) , and powers in pure form stand up at the base of functional . Collatz Tree is a Math beauty and come from , or revealed by a simple Conjecture also , which give my the right to name it , Collatz-Murgu Conjecture expresed by
{1 I +1 }
and follow the same rule only that 3 , now is 1.
For sure the tree will not have the same beauty and extension , but will be a Step as logical explanation for Collatz one. But also we have near as beauty another two Murgu - Collatz Conjecture which this time posses everyone any proper beauty and genertaed by :
1. [2*I + 2 ] following the same rules .
2. [2*I - 2 ] following the same rules .
but also not proved yet anothers 2 , tested partialy only :
1. [4*I + 4 ] following the same rules .
1. [4*I - 4 ] following the same rules .
A beauty which need a rest time attention, is which I named Collatz-Murgu Magic Circles .
or any contest any - Murgu Magic Vicious Circles
For Collatz Generator I change a sign on
[3 I - 1] , and then as will see if for Collatz Conjecture consider 1 (Unity as root) as root , for Murgu Vicious Conjecture apear any multiple roots as 1, 5, 17 , .. and the for Integer 17 we get :

17	50
	25	74
		37	110
			55	164
				82
				41	122
					61	182
						91	272
							136
							68
							34
							17

Maybe Corect name for this property is Vicious Circle.
But as a Math Beauty this Conjecture come with a strange behavior. which can be named as:
Vicious Conjectures.
To resume it with any tables as examples, where will to not extra fullfill the space will follow only odds numbers of nodes starting from biggest one.

	3	1
	5	7	5
	7	5	7	5
	9	13	19	7	5
	11	1
	15	11	1
	23	17	25	37	55	41	61	91	17
	29	43	1
	31	23	17
	67	25	37	55	41	61	91	17

23 , 31 , 41, 37,55, 61, 67, 91 fall also into vicious circle of 17.

71	53	79	59	11	1
73	109	163	61	91	17
111	83	31	23	17
139	13	19	7	5
157	235	11	1

7921	11881	17821	26731	1253	1879	1409	2113	3169	4753	7129	10693	16039	12029	18043	3383	2537	3805	5707	535	401	601	901	1351	1013	1519	1139	427	5
7923	2971	557	835	313	469	703	527	395	37	55	41	61	91	17
3779	1417	2125	3187	1195	11	7	5
3119	2339	877	1315	493	739	277	415	311	233	349	523	49	73	109	163	61	91	17

Murgu Conjectures Vicious Redundancy became provocative. I come with any example which proof a subtle play between roots (1,5,17,,) and I was lucking to get also another one. Not luck!
Root 5 as a "leaf bud" and root 17 as "leaf" in a tree took My attention, and also I Tried to getone more even for sure will be far away or even will be a "Leaf Explosion" (diameter evolving to infinity).
Those new properties maybe can be used into Cryptography or Graph Theory.
To express redundundancy for , I named also those leafs as Magic Redundant Circles>
Those don't bring to much for Logic of Science, then maybe any for Conventional Science, which is more also as gaining quality.

Murgu Conjecture Vicious REDUNDANCY

9377	14065	21097	31645	47467	2225	3337	5005	7507	2815	2111	1583	1187	445	667	125	187	35	13	19	7	5
19937	29905	44857	67285	100927	75695	56771	21289	31933	47899	8981	13471	10103	7577	11365	17047	12785	19177	28765	43147	4045	6067	2275	853	1279	959	719	539	101	151	113	169	253	379	71	53	79	59	11	1
37199	27899	5231	3923	1471	1103	827	155	29	43	1
39119	29339	5501	8251	1547	145	217	325	487	365	547	205	307	115	43	1
71993	107989	161983	121487	91115	4271	3203	1201	1801	2701	4051	1519	1139	427	5
91193	136789	205183	153887	115415	86561	129841	194761	292141	438211	164329	246493	369739	34663	25997	38995	14623	10967	8225	12337	18505	27757	41635	15613	23419	4391	3293	4939	463	347	65	97	145	217	325	487	365	547	205	307	115	43	1
91195	17099	1603	601	901	1351	1013	1519	1139	427	5
91197	136795	25649	38473	57709	86563	32461	48691	18259	6847	5135	3851	361	541	811	19	7	5
199933	299899	56231	42173	63259	11861	17791	13343	10007	7505	11257	16885	25327	18995	7123	2671	2003	751	563	211	79	59	11	1

1383961	2075941	3113911	2335433	3503149	5254723	1970521	2955781	4433671	3325253	4987879	3740909	5611363	2104261	3156391	2367293	3550939	665801	998701	1498051	561769	842653	1263979	59249	88873	133309	199963	37493	56239	42179	15817	23725	35587	13345	20017	30025	45037	67555	25333	37999	28499	10687	8015	6011	1127	845	1267	475	89	133	199	149	223	167	125	187	35	13	19	7	5

Murgu Conjecture Vicious REDUNDANCY come with a maybe Big Mathematics Curiosity , are x=5 and x=17 singles "roots" or we have far away another ? Are leafs exploding into infinity already and bring an oscilating property? Are pure curiosity only.

BUT now if we will connect Murgu Conjecture Vicious REDUNDANCY whith Colatz Conjecture we will get , maybe or not, future will say a nice property for Numbers Theory and also maybe the best solution for Collatz conjecture .

For every Integer L >1, L≠(3∗i) which can be written as:
1a. (2^L + 1 = 3 x)
2a. ((2^L - 1 = 3 x))
only one is TRUE time by time as 1a. for L odd and 2a. for L even .
This is now testable for millions of L's , and is no motivation to not be true to infinity, but a accurate mathematics demonstration may be a chalenge for future.
For Numbers Theory this may be a good Numbers property.

This prov Collatz and Murgu Conjecture Vicious REDUNDANCY are spreaded to infinity , but for Murgu Conjecture Vicious REDUNDANCY can't solve "leaf" aspects and if will can have more "roots" .
Also for Collatz and Murgu Conjecture Vicious REDUNDANCY this don't cover all Integers , but only mirroring it to infinity. Then to make any work for :

For it I will define 6 Murgu-Collatz Conjectures which do not need a demonstration
MC1: (x+1); following Collatz rules.
MC2: (x-1); following Collatz rules.
MC3: (2x+2); following Collatz rules.
MC4: (2x-2); following Collatz rules.
MC5: (4x+4); following Collatz rules.
MC6: (4x-4); following Collatz rules.
With all clearity MC1. and MC2. cover all Integers , when the rest , even if do it,apparent need a demonstration. MC1 and MC2 will be evry time evolving to a close 2^L then never will fall .
MC3 and MC4 are followers of those in the front or in the back creating a dizzy Infinity Paradox in our minds , but then with aacuracy play in the same mode only wearing differents steps . The same for MC5 and MC6 .
If will notate Collatz Conjecture and Murgu Conjecture Vicious REDUNDANCY as :
MC7: (3x+1); following Collatz rules.
MC8: (3x-1); following Collatz rules.
First , I will remind to you Double False Redundancy Of Truth as to remark Multiply can brought any new , to not say Quality , but maybe property. Then by a multiply with 2 and for of those:
MC9: (6x+2); following Collatz rules.
MC10: (6x-2); following Collatz rules.
MC11: (12x+4); following Collatz rules.
MC12: (12x-4); following Collatz rules.
Maybe MCC11 and MC12 will fall faster into a infinty explozion , but at the last MC9 and MC10 will not .
Normaly for any like (Nx + 1), like I said before , a mirror Collatz conjecture usually fall because N > 4 , contain 2^K and K > 1 can to erode the meting with a 2^L. Murgu Conjectures seem then to bring new asspects to Collatz Conjecture and Murgu Conjecture Vicious REDUNDANCY.
Imporatante maybe only for Graph Theory and Cryptography , but nobody can't to say if in Time will not reveal also any new Numbers Theory Prperties .
Which is interesting MC10 is following with respect Murgu Conjecture Vicious REDUNDANCY property at the last for 5 and 17, I alreday tested .

17	100
	50
	25	148
		74
		37	220
			110
			55	328
				164
				82
				41	244
					122
					61	364
						182
						91
						91	544
							272
							136
							68
							34
							17

The Effect is the same but if odd numbers are the same, even numbers gain for every step one more division with 2. Isn't a paradox but we can see is a fantasy.
I worked a bit , then to name it
"Murgu Conjectures Fantasy".
I am sorry, I am not so rich to pay for Speciality Magazines money to publish those , but
Climatic Disorder DOTCOM
is for a long time hosting a lot of Science and specialy
Humanity Science And Culture Universal Thesaurus
best components
America Earth Proud Day Fundamentals
or USA AEPDF
for which I am sure will influence a lot Science Future .

For a Rest_Time time we get any , but now for a while I want to present to you any for future in Math!

Fermat_Murgu Quadruplets for Z< 902

Important Murgu Quadruplets Generators.

— Z  —	— X  —	— Y  —	Fmin X,Y,Z
9	8	6	1
12	10	9	-1
103	94	64	-1
144	71	138	1
150	73	144	-1
172	138	135	1
249	235	135	-1
495	438	334	-1
505	426	372	1
577	486	426	1
729	242	720	1
738	244	729	-1

Thanks and Good _Morning Future! for all.