Fermat couldn't possibly have had this proof. - Andrew Wiles

I hope that seeing the excitement of solving this problem will make young mathematicians realize that there are lots and lots of other problems in mathematics which are going to be just as challenging in the future.

However impenetrable it seems, if you don't try it, then you can never do it.

But what has made this problem special for amateurs is that there's a tiny possibility that there does exist an elegant 17th-century proof.