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
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