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''.