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 exsist. I was hoping to find a proof that would look more ''Fermatian''