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Topic: Fermat's last theorem
Replies: 16   Last Post: Nov 11, 2017 7:33 AM

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Jarno grenson

Posts: 12
From: Belgium
Registered: 10/20/17
Fermat's last theorem
Posted: Oct 20, 2017 7:33 AM
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Hey guys, i'm new here and i would like to get some input about a possible proof of fermat's last theorem. I do think i didnt make any mistakes, but i would like to see wether i actually did. Here is it:

We have
A^n+B^n = C^n --(1)

We state that
B<A<C<A+B, --(2)
gcd(a,b) = gcd(a,c) = gcd(b,c) = 1
All numbers are part of N

And we assume that
n is odd and bigger than 2
C = (P^k)*D with P prime

We rewrite (1) as

(E)^n + (F)^n = P^(kn)

 Where E and F are A/D and B/D respectively
We can rewrite the left side as

(E+F)* (E^(n-1)-E^(n-2)*F +... -E*F^(n-2)+F^(n-1))

We know from (2) that
P^k<E+F< 2*P^k

This also means however that the other part, which we replace as S, holds true to these conditions:
P^(kn-k)÷2 < S < P^(kn-k)

These inequalities mean that (E+F) can be a multiple of P^(k-1) at most, while S can be a multiple of P^(nk-k-1) at most.

This means however that the product of those two parts can only be a pruduct of P^(nk-2) at most, since P is prime. We see a clear contradiction, so we conclude that there doesnt exist an odd number n where the equation is true.

This coupled with the proof of  fermat himself for n=4, proves that there does not exist any n bigger than 2 where the equation is true

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