Yeah, mine too. I had a professor who showed me a Xerox of a test where the question gave a proposition, and asked you to either prove it or give a counterexample. Of course, this guy had done both. 
I believe most mathematical historian types agree with this. It seems likely that Fermat’s mistake was assuming that you could extend the integers by a root of unity and still have unique factorization like you do with regular integers. Unfortunately, I’m already casting things in terms of modern mathematics. I’m not sure how to describe how Fermat would have phrased things.
It actually speaks pretty well of Fermat to say that he realized his proof was lacking. For one thing, he was chugging along centuries before modern algebra. For another, this proof would work for n<23. For a third thing, later mathematicians seem to have made the same mistake even when it was known that extensions to Z might not have unique factorization.
For the OP I’ll chime in my support for the Reimann hypothesis. I suppose I can’t think of a problem as nice as Fermat’s for being simple to state and for having a lot of interesting ways to attack it.