FERMAT LAST THEOREM:ANOTHER PROOF

From: george ghiata (george_ghiata_at_hotmail.com)
Date: 01/13/05 

Date: Thu, 13 Jan 2005 02:35:14 +0000 (UTC)

This is a proof of Fermat Last Theorem by Gheorghe Ghiata
EQF: X^+Y^n=Z^n (X,Y,Z Are integers and n>=3) is impossible
PROOF:
   X^n-X+Y^n-Y=Z^n-Z+Z-X-Y
  X+Y-Z=B; Little theorem makes B divisible n
Case1:X*Y*Z not divisible by n
  X=B+Q
  Y=B+P
  Z=B+Q+P
  EQ1 :(B+Q)^n+(B+P)^n=(B+Q+P)^n
 From EQ1 we get:
  X+Y=2*B+Q+P=U^n=W
   Z=U*V
   Q=q^n and P=p^n
Therefore 2*B=U^n-q^n-p^n
Little Theorem makes (U-q-p) divisible by n:
Now we write:
    X^n-Q^n+Y^n-P^n-Z^n +W^n=(-Q^n+Q)+(-P^n+P)+(W^n-W)+
                                                       +(W-Q-P)
    Therefore (W-Q-P)=2*B is divisible by n^2
Therefore B is divisible by n^2
Now EQ1 is divided by (2*B+Q+P) and get
 EQ2:
[B^(n-1)+(a1)*B^(n-2)+(a2)*B^(n-3).........+c*B]+[Q^n+P^n]/(Q+P)=V^n
 From EQ2 we get that (V-1) is divisible by n and that
        {V^n - [Q^n+P^N]/(Q+P)} is divisible by n^2
If we divide EQF by (X+Y) we get :
 EQ3: X^(n-1)-Y*X^(n-2)+[Y^2}*X^(n-3)..........+Y^(n-1)=V^n
After we multiply EQ3 With Z we write It in two ways:
X+Y=Z+B=W
 1.
Z*{W^(n-1)-n*[W^(n-2)]*Y+(1/2)*n*(n-1)*[W^(n-3)]*Y^2........+n*Y^(n-1)=Z*V^n

 2. The same as 1. exept we substitute Y with X
  Now we Add -up (1.+2.) and get:
  EQ4: B*G +(Z-1)*n*[Y^(n-1)+X^(n-1)] +Q^n+P^n+X^n+Y^n=2*Z*V^n
  Now we substitute {X^n+Y^n] with Z^n and get to the left side of
EQ4
the value 2*Z*V^n.Since B is divisible by n^2 we get that (Z-1) is
divisible by n.
But Z=U*V and (V-1) is divisible by n.
Therefore (U-1) is divisible by n.
Repeating the proof twice we get That (q-1) and (p-1) are divisible by
n too.
But We know that (u-q-p) must be divisible by n too.
That proves Fermat last theorem for Case1.
Case 2; Z is divisible by n
In this case ((q+p) is divisible by n.
As in Case 1. we proof that (q-1) and (p-1) are divisible by n.
Therefore this proves Case2 of Fermat Last theorem
Therfore Fermat Last theorem Is true.
Created by Gheorghe Ghiata

                  

  

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