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by generalizing over all n's as powers make possible
WITH ALL SCIENCE RIGOR AND CLEARITY as -

Begin with 2015 September 24
- Fermat's Last Theorem - FUNDAMENTAL.

Begin with Earth Proud Day - September 24 2015 - FERMAT LAST THEOREM IS FUNDAMENTAL IN SCIENCE WITH ALL ABSOLUTE RIGHTS AND IT WAS POSSIBLE BECAUSE OF:
1.ION MURGU - INTEGERS POWERS FUNDAMENTAL EQUATIONS.
2. FERMAT - MURGU IMPOSSIBLE EQUATIONS - which maybe are improper named, but is a perfect Mathematics Method, which with accuracy, and all science rigor SENT Fermat's Last Theorem in FUNDAMENTAL - INCLUDING ALSO Pythagorean Triples as BLESSED EXCEPTIONS which confirm the rule. FERMAT - MURGU IMPOSSIBLE EQUATIONS generalize over all n

Fermat - Murgu Impossible Equations - an Analytic Math Method for Fermat's Last Theorem Analyze. It started by using Ion Murgu Integers Powers Fundamental Equations , proved for n=1 to n=50, which I supposed to be enough for Generalize(maybe soon will be extended to 70, which are Powers with which usually rarely work). ALSO THIS METHOD IS THE SINGLE METHOD WHICH CAN CERTIFY Fermat's Last Theorem in an accurate mode, with clearity, and Math Severity. There now are another 2 Methods as , Euler - Murgu Equations 1=1 and Fermat's Last Theorem Natural Solution, but both are starting , coming, from FMIE WHICH IS A COMPLETE METHOD. I named it so, because reveal The impossibility for n>2 as Xn + Yn − Zn = 0 to have any solutions into Integers . For it we applied the property of additivity of Fermat-Murgu n Media to (X,Y,Z), (X-1,Y-1,Z-1) and we can do it for (n+1)/2 into the left side and the same to the right, but for our scope will be enough one to left and its proper one. Doing it for (X,Y,Z), we get Fermat-Murgu First Grade Impossible Equations which reveal as, Fermat Equations to have all solutions into integers , then, Fermat-Murgu n Media for 3 Integers associated will be unbalanced .

To understand Fermat - Murgu Impossible Equations, need to SEE Ion Murgu Integers Powers Fundamental Equations and to play a litle bit with.

Ion Murgu Integers Powers Fundamental Equations PROOF »

and now also in a C Sharp program, here at HST_Proof

And Formula at:

Ion Murgu Integers Powers Fundamental Equations Formula »

  0
  ∑ [ (-1)m(kIn)(T+I)n]= n!
  I=n

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

To Start The Analyze with {X,Y,Z } itself.

For this Scope to work by now with Fermats Last Theorem General form as Fermat Equations ;
Xn + Yn = Zn
and for easy work to write it also as
Xn + Yn - Zn = 0

To asociate to it 3 Integers using IMIPFE :
  0
  ∑ [ (-1)m(kIn)((X+I)n + (Y+I)n - (Z+I)n)]= n!
  I=n

FMIE Asssociate to Fermat Equations Integers
THEN, IF :
Xn + Yn - Zn = 0

then
  1
  ∑ [ (-1)m(kIn)((X+I)n + (Y+I)n - (Z+I)n)]= n!
  I=n

Fermat - Murgu First Grade Impossible Equations
to remark we loss last therm , and the rule is Damaged. Fermat - Murgu n Media ,Fermat Equations associated, for 3 Integers is:
  0
  ∑ [ (-1)m(kIn)((X+I)n + (Y+I)n - (Z+I)n)]= n!
  I=n
and can't be unbalanced by any Zero Therms , but we can't Postulate it, and then to say by now:

Fermat - Murgu First Grade Impossible Equations brought first PROOF,Fermat was right, Fermat-Murgu n Media UNBALANCED, but we can't postulate Fermat-Murgu n Media (associate) RULES even if intuitively is clear- CAN"T be unbalanced.

Fermat Equations to have any solutions into Integers , Fermat-Murgu n Media need to be UNBALANCED, and its Last Term to be ZERO, and to Analyze the next one.

To start an Analyze for a neighbors, for {(X-1),(Y-1),(Z-1)}
  0
  ∑ [ (-1)m(kIn)((X+I-1)n + (Y+I-1)n - (Z+I-1)n)]= n!
  I=n
Second FMIE Asssociate to Fermat Equations Integers
IF :
Xn + Yn - Zn = 0

Then:
  2
  ∑ [ (-1)m(kIn)((X+I)n + (Y+I)n - (Z+I)n)]= n!(+/-) [(X-1)n + (Y-1)n - (Z-1)n ]
  I=n
Fermat - Murgu Second Grade Impossible Equations
Equations wich SENT Fermat's Last Theorem into FUNDAMENTAL.

HOW SO ?

Forgot for a while, even if we used it :

Xn + Yn - Zn = 0

was a supposition. Then:

buiding for Fermat - Murgu Second Grade Impossible Equations it's matrice form, and solving it. Then will get:

n(Xn + Yn - Zn) = 0
which claim Irrational solution and to remark also , not complex solutions,
which make now Euler - Murgu Equations 1=1 a Method of SOLVING Fermat's Last Theorem.
But the ACCURACY WAS from FMIE.
(X)n√ n   ,  (Y)n√ n   ,  (Z)n√ n  

To handle, for a while, our appetite to Simplify, and to re logic any . For Equations like Fermat-Equations we know :

Xn + Yn - Zn = 0

Solution is - {X,Y,Z} but really can be {IX,IY,IZ} and when connected to Physics Processes we need to determine which is real one.

Also to not forgot , we get

n(Xn + Yn - Zn) = 0

via a corect calculus , even if also intuitivelly been waiting for is a corect intuition with all rights.


With all Rights

n(Xn + Yn - Zn) = 0

ABSOLUTE CONDITIONAL EQUATIONS WHICH SENT Fermat's Last Theorem in FUNDAMENTAL By Sending for all n>1 Fermat Equations Base Solutions into IRRATIONAL FIELD and Revealing Pythagorean Triples as Blessed Exception which obey to the Rules and Then Confirm The Rule. THOSE CONDITIONALS ARE COMING FROM A CORECT CALCULUS, AND THEN THE RULES OBLIGE TO RESPECT THEM IN A FORM WE GET IT.Solution are of form:

(X)n√ n   ,  (Y)n√ n   ,  (Z)n√ n  

THOSE AREN'T REAL SOLUTIONS FOR FERMAT'S LAST THEOREM ADRESED TO INTEGERS, but now we know that:
Fermat's Last Theorem have solutions in IRRATIONALS, without a possible flow to Rationals and Integers and Inverse sense, which for Exceptions Pythagorean Triples the flow is clear.

Every Integer Z at every power n>1 Zn, can't be write as two terms integers sum at the same power n, but instead can be write as a sum of two Irrationals at power n connected by theirs image in unity as Irrationals Complementary to Unity, then infinity.
For it see Euler - Murgu Equations 1 = 1.

If any doubts about - SPECIAL IRRATIONAL (n√ n) ,SEE MURGU IRRATIONALS



By mathematical Perception, even simple and pure association a Fermat Eqauations to The Lost Fundamental , exclude Fermat's Last Theorem into Integers.

  0
  ∑ [ (-1)m(kIn)((X+I)n + (Y+I)n - (Z+I)n)]= n!
  I=n

But by a stricte Mathemathical Analyze of symetric n/2 neighbors , we get Fermat - Murgu Impossible Equations
[n(Xn + Yn - Zn) = 0]
[(n - 1)(Xn + Yn - Zn) = 0] [(n + 1)(Xn + Yn - Zn) = 0]
[(n - 2)(Xn + Yn - Zn) = 0]
etc . (n√ n) special irrational exclude Xn + Yn - Zn = 0 into INTEGERS AND RATIONALS.
That mean logic we Solved Fermat's Last Theorem by a
ABSOLUTE CONDITIONAL
for every power n, as [Zn = Xn + Yn] is forced to satisfy:


  n/2 n/2
(∏ (n - J)) (∏ (n + J)) (Zn - Xn - Yn) = 0
J = 1  J=1

This isn't nothing more then a CONDITIONAL.
I wrote those Equations conditional using a wrong temporal memory, and I realy hope isn't in 'Infinity Divergent Conjecture.pdf' also and also to be pardonable. We can speak here about a produce of n/2 terms (kIn), but like said we didn't get not Fermat's Last Theorem SOLUTIONS here , but an absolute truth CONDITIONAL which say Fermat was right. For every Fermat Equation
Xn + Yn = Zn - SOLUTIONS -
one, or two or all X,Y,Z must to be IRRATIONAL. As example for n=3
3 + 4 = 7 -> X = 3√3 ; Y = 3√4 ; Z = 3√7
8 + 19 = 27 -> X = 3√8 ; Y = 3√19 ; Z = 3√27
9 + 18 = 27 -> X = 3√9; Y = 3√18; Z = 3√27


Mathematics is a beauty!
but any times Physics is driving it in good directions by sonding its logic.
If you put a break to simplify old instincte you will see a nice combinations of irrationals which will not lose THE SENSE even for power n -> ∞ , and luck at SPECIAL IRRATIONALS DEMONSTRATION FOR.


Begin with 2015 September 24
FERMAT'S LAST THEOREM IS FUNDAMENTAL IN SCIENCE

America Erth Proud Day Fundamentals

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