We just don't know whether there exists a 'short cut' proof (or, as it is sometimes called, a proof 'from the book') at all that only uses 17th century maths solving this question. As Andrew Wiles said himself, his proof was surely 20th century, so Fermat would not have had that proof. But maybe Fermat had a proof of his own, or maybe Fermat was wrong in thinking he had a proof. What is known, is that he left behind some dazzling examples of factorization that are hard to explain with 17th century tools, so he surely had capabilities...it makes the thought he just might have had a short proof even more tantalizing.
Interesting even more is that this question on whether shorter proofs always exist (like the transcendence of pi by Hilbert, or the Prime Number Theorem by Erdos) look a lot like the P vs NP question, on whether problems that we now only can easily verify the answer of, are actually all easy problems, had mankind just been smart enough to just see these easy solutions! Maybe Wiles' proof of Fermat is as short as it gets, and if so, Fermat must have been wrong.

We just don't know whether there exists a 'short cut' proof (or, as it is sometimes called, a proof 'from the book') at all that only uses 17th century maths solving this question. As Andrew Wiles said himself, his proof was surely 20th century, so Fermat would not have had that proof. But maybe Fermat had a proof of his own, or maybe Fermat was wrong in thinking he had a proof. What is known, is that he left behind some dazzling examples of factorization that are hard to explain with 17th century tools, so he surely had capabilities...it makes the thought he just might have had a short proof even more tantalizing.

Interesting even more is that this question on whether shorter proofs always exist (like the transcendence of pi by Hilbert, or the Prime Number Theorem by Erdos) look a lot like the P vs NP question, on whether problems that we now only can easily verify the answer of, are actually all easy problems, had mankind just been smart enough to just see these easy solutions! Maybe Wiles' proof of Fermat is as short as it gets, and if so, Fermat must have been wrong.