Q:

Fermat's proof for $x^3-y^2=2$

Fermat proved that $x^3-y^2=2$ has only one solution $(x,y)=(3,5)$.
After some search,i found only proofs using factorization over the ring $Z[\sqrt{-2}]$.
My question is: 
Is this Fermat's original proof?If not where can i find it?
Thank you for viewing
Note: I am not expecting to find Fermat's handwritings  because this may not exist.
I was hoping to find a proof that would look more ''Fermatian''

A:

Fermat never gave a proof, only announced he had one (sounds familiar?). Euler did give a proof, which was flawed, see Franz Lemmermeyer's lecture notes, or see page 4 of David Cox's introduction.
For a discussion why a proof along the lines set out by Fermat is unlikely to work, see this MO posting.
---- trivia ----
As a curiosity, I looked up Fermat's original text (reproduced below from his collected works), written in the margin of the Arithmetica of Diophantus:

Can one find in whole numbers a square different from 25 that, when
  increased by 2, becomes a cube? This would seem at first to be
  difficult to discuss; and yet, I can proof by a rigorous demonstration
  that 25 is the only integer square that is less than a cube by
  two units. For rationals, the method of Bachet would provide an infinity
  of such squares, but the theory of integer numbers, which is very 
  beautiful and subtle, was not known previously, neither by Bachet,
  nor by any author whose work I have read.

A:

Fermat did not prove this result; he claimed that the only solution is the obvious one and conjectured (in words that seem to suggest he knew how to prove it, but without explicitly saying so) that this can be proved by descent. I am sure that Fermat, if he really believed 
to have a proof (in my opinion he did not), was mistaken.
I am not aware of any proofs based on Fermat's techniques alone, and I have often tried
to find one myself - so far without success.

