Fermat proved that $x^3-y^2=2$ has only one solution $(x,y)=(3,5)$.


After some search, I only found 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 they may not exist.
I was hoping to find a proof that would look more ''Fermatian''.



Fermat's proof for $x^3-y^2=2$
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,](http://www.fen.bilkent.edu.tr/~franz/ant/ant01.pdf) or see page 4 of David Cox's [introduction.](http://math.stanford.edu/~lekheng/flt/cox.pdf)


For a discussion why a proof along the lines set out by Fermat is unlikely to work, see this [MO posting](http://mathoverflow.net/questions/36800/fermats-bachet-mordell-equation).


---- 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.
> 
> 
> 


![](https://www.lorentz.leidenuniv.nl/beenakker/MO/Fermat_margin_note.png)


