Murgu The Lost Fundamental


Murgu The Lost Fundamental
Unknow Until 2015
As All USA_AEPDF: Knowledge Labyrinth Extraction
Natural Placed Itself at The Base Of Science Base
and for sure will get rights of Operator in High Level Science
The Lost Fundamental Operator


AMERICA EARTH PROUD DAY FUNDAMENTALS
Component

Under Humanity Natural Law Of Copyright


and USA ISBN-978-1-63972-043-9


by and before 2015 September 24


free for Education and Science Students as .


USA AEPDF
America Earth Proud Day Fundamentals
Murgu AEPDF or Murgu AEPDF 11 including
Results from, all are
UNKNOWN SCIENCE UNTIL 2015
Knowledge Labyrinth Extraction
, UNDER
USA Intellectual Property



Started from Analytic study of Powers to Discover any hide Properties which return real importance for all Science as Multiply, Divisions and then Powers stand at base of Math Science and even Science.
Result was published on Internet , first as
Math Millennium Equations
then soon as
Mathematics Millennium Equations
and after any time becaming aware about real importance
Integers Powers Fundamental Equations.
But those equations are fundamentals which placed themself at the base of Fundamental Science, then can be named
THE LOST FUNDAMENTAL .
and with you accept and to strong Humanity Human Legacy(at the last as Intelligence)
Murgu - THE LOST FUNDAMENTAL .
How Important are?
Near of Instantly, using adequate Math Method, Fermat Murgu Impossible Equations , SENT
Fermat's Last Theorem - FUNDAMENTAL
lightening near of 400 years of research for - Fermat's Last Theorem and with tools resulted from and your accepted brougth
Murgu - Fermat's Last Theorem Addition.

THE LOST FUNDAMENTAL

or Ion Murgu - Integers Powers Fundamental Equations
had first aproach around of '75-'76 when calculus error, avoided the chance to be Discovered then. First Method for geting Proof for existence, will be Explained Down.
This started using a intuitive schema for general case, but then I worked directly with powers, testing it for particular cases for n=2,n=3,n=4,n=5,n=6 as powers. Then, I make any calculus mistake for any powers, and was thinking schema isn't validated for general cases.
But in early of 2015 I returned to calcul for the same powers and noticed My old Table Schema was answering positive to may intuition.
Down will present it and 2 Example for powers 4 and 5. (I did it also for 6 and 7 )



Tn (T+1)n (T+2)n (T+n-1)n (T+n)n
(T+1)n-Tn (T+2)n-(T+1)n (T+3)n-(T+2)n (T+n)n-(T+n-1)n
dif. of superior terms dif. of superior terms dif. of superior terms
dif. of superior terms dif. of superior terms
n!

Working Table Schema.


and calcul now for n=4 , starting from T=2

16812566251296
65175369671
110194302
84108
24

24 is also 4!
Power 4 Example.



 3125      7776      16807      32768      59049      100000                   
    4651      9031      15961      26281      40951                         
4380 6930 10320 14670
2550 3390 4350
840 960
120

Power 5 Example.
120 is also 5!.


GETTING FORMULA


What been waiting for, is proved, as remark I did it for n=2, n=3, n=4, n=5, n=6 and n=7, then not a motivation to exclude general case, and late on I make a software which proved it for n between 1 and 50.
Now is the time to remark
Murgu - The Lost Fundamental is IMPORTANT FOR SCIENCE
for Powers 2, 3 and maybe 4 and maybe into the future Power 5 also. This isn't for calcul powers. But Math will can use it in polinomials for superior powers also.
This started from Physics and get important isues for. The Time will prov it.
For Mathematics proofs are only provocations for extracting any formulas. Then, I made second STEP, to get The Formula, and for it I keept the powers from first row as functions of I and n. We will do it for n=4 and 3(I did also for 5).
For general case is heavy to make a table schema, as there will appear Coeficints which depend of n as power but also by position in formula.
But, for every Power n is a Mathematic Symmetry which make it easy to get every coeficient. I make a Software ( a PERL program to get coefints for any power n, and one to proof future Formula for powers 2 to 50).
The Lost Fundamental Formula have importance for Science for Powers 2, 3 and maybe 4, but never know if into future will not evolve to 5,6 and 7.


fTfT+1 fT+2 fT+3 fT+4
fT+1-fT fT+2-fT+1 fT+3-fT+2 fT+4-fT+3
fT+2-2*fT+1+fTfT+3-2*fT+2+fT+1fT+4-2*fT+3+fT+2
fT+3-3*fT+2+3*fT+1-fT fT+4-3*fT+3+3*fT+2-fT+1
fT+4-4*fT+3+6*fT+2-4*fT+1+fT

Table Explicative for power 4, how to get Formula.
Remark apparition of coeficients to last.


 T3      (T+1)3     (T+2)3      (T+3)3    
    (T+1)3-(T)3     (T+2)3-(T+1)3      (T+3)3-(T+2)3       
       (T+2)3-2(T+1)3+(T)3      (T+3)3-2(T+2)3+(T+1)3           
          (T+3)3-3(T+2)3+3(T+1)3-(T)3             

Table Explicative for power 3, how to get Formula.
Remark apparition of coeficients to last.



For general cases a Table will be To big , but an atempt can be made(I did) as needed to see the Evolving Coefficients (seem to be a symmetric doubled Pascal Triangle).
Remind , Pascal Triangle was get from another Math Issue.
A Computing form is a easy procedure, because every superiors Coefficients are evolving also from the precedents.
Anyway , using above Table I get Murgu- The Lost Fundamental Formula for General Cases, for which correct name is
Murgu- The Lost Fundamental Equations
how is different for every power.



Murgu The Lost Fundamental

n=111
n=2121
n=3133 1
n=41464 1
n=515101051
n=61615201561
n=7172135352171
n=818285670562881
n=9 1 9 36 84 126 126 84 36 9 1
n=10 1 10 45 120 210 252 210 120 45 10 1
n=11 1 11 55 165 330 462 462 330 165 55 11 1
n=12 1 12 66 220 495 792 924 792 495 220 66 12 1
n=13 1 13 78 286 715 1287 1716 1287 715 286 78 13 1
n=1411491364100120023003343230032002100136491141
n=1511510545513653003500564356435500530031365455105151
n=16116120560182043688008114401287011440800843681820560120161
n=17117136680238061881237619448243102431019448 1237661882380680136171
n=18118153816306085681856431824437584862043758318241856485683060816153181
n=191191719693876116282713250388755829237892378755825038827132116283876969171191
n=201201901140484515504387607752012597016796018475616796012597077520387601550448451140190201
n=21121210133059852034954264116280203490293930352716352716293930203490116280542642034959851330210211

Power 2 To 21 - coeeficients table -
To show until poer 50 will make heavy this page.


Now we have Formula in the front of us, but also can be write it in two forms wich will bring in the front SIGN Problem for wich every Mathematician know how to deal. Strict decision about need maybe International meeting.



Form 1

  0
  ∑ [ (-1)m(kI,n)(T+I)n]= n!
  I=n

WHERE m IS ; FOR N ODD m=(I+1) , FOR N EVEN (m=I)



Form 2, inverse descend

  0
  ∑ [ (-1)m(kI,n)(T-I)n]= n!
  I=n

WHERE m IS ; FOR N ODD m=(I+1) , FOR N EVEN (m=I)



To avoid Sign Problem Convention , Murgu - Thel Lost Fundamental Equations General Formula can be write as:



Murgu The Lost Fundamental

THOSE INFINITY TO INFINITY EQUATION - IDENTITIES ARE IMPORTANT FOR , BECAUSE EXPRES ABSOLUTE TRUTH . EVEN PRIME NUMBERS GET NOW THE CONNECTION WITH ALL NUMBERS ON.
THOSE INFINITY TO INFINITY EQUATION - IDENTITIES
been obtained from Knowledge Labyrinth as extraction from, and
I have dare, with your permission, to say , are maybe TOP1 Of
- HUMANITY SCIENCE THESAURUS -
. And it, not because I get it, but because are. Are simple now, but we needed 7000 years of Science to get it. To not confuse it whith Pascal Equations what been obtained by simple evolving of (a+b)n by multiple multiply of (a+b)... (a+b). Here we get the real conection whth n!.
In first place, I named THOSE ,
Murgu Math Millennium EQUATIONS
soon after
Murgu - Mathematics Millennium Equations -
and , aware of theirs Math role
ION MURGU - INTEGERS POWERS FUNDAMENTAL EQUATIONS
but started from Physics and will play a big role in all Science, named those


MURGU - THE LOST FUNDAMENTAL
Those Equations are Eternal Fundamental Science Absolute Truth Values (EFSATV).

TO REMARK the property of adition and then WE CAN GET NOW Infinity to Infinity Connection between all Integers Including Prime Numbers Prime Numbers - a GOOD NEW TOOL into Numbers Theory, but for sure also into work with pollinomial functions.


Ion Murgu- Integers Powers Equqtions,for sure, will play a role in Numbers Theory, but My hope is to help Science to extract from Knowledge Labyrinth more
Truth Science Values.
A Perl Program can be helpful in convincing one more time. See one of to links for!


2015 Science The Lost Fundamental name were changed as
Murgu The Lost Fundamental
to avoid confusion with The Lost Fundamental from Golf.
Murgu The Lost Fundamental was know also as
Math Millennium Equation (first), Mathematics Millennium Equations and Integers Powers Fundamental Equations.
and myself consider it best Human Discovery which is Knowledge Labyrinth Extraction.
In Time Science will decide if I was right, remarking, Math Science implication begin via Multiplay, Division, Powers.
Those stand natural at the base of Science.