I recently came across the following passage which is about Fermat's Last Theorem (FLT).

*Pierre de Fermat had found a proof, but he did not bother to write it down. This is perhaps the most frustrating note in the history of mathematics, particularly as Fermat took his secret to the grave.*
AH- so at one time people thought Fermat DID have a proof of FLT. That is, a proof using just the math of his time, likely a very clever proof. I doubt anyone thinks that Fermat had a proof in this day and age. Actually it has been in fiction: in a 2010 episode Dr. Who episode

*The eleventh hour, *the doctor has to prove to some very smart people that they should take his advice. He does this by showing them Fermat's proof of FLT. Good Science Fiction but highly unlikely as Science Fact. In an episode of ST-TNG (Title: The Royale. Year: 1989) it is claimed that FLT is still open. Whoops. But in an episode of ST-DSN (Title: Facets. Year: 1995) they refer to `Wiles proof of FLT'.

Wikipedia states:

* It is not known whether Fermat had actually found a valid proof for all exponents n, but it appears unlikely.* I think that understates the case.

So here is a question for all you math historians out there: When did the math community realize that FLT was really hard?

We have one clue- the quote I began with. Its from... 2013. Whoops. The book is

*The Simpsons and their mathematical secrets *by Simon Singh (Author of

Fermat's Enigma which is about the quest to proof FLT). I've read the passages about FLT in the Simpsons book over again to make sure he doesn't someplace say that Fermat prob didn't have a proof. No--- he seems to really say the Fermat had a proof. So whats going on here? Possibilities:

1) I'm wrong. There are serious credible people who think Fermat had a proof and he talked to them; perhaps while working Fermat's Enigma. I find this unlikely- I have never, literally never, heard of anyone, not even math cranks, who think Fermat had a simple proof. Some cranks think THEY have a simple proof, though even that seems far less common after FLT was proven.

2) I'm right. He didn't have anyone who was both serious and credible check his book. I find this unlikely. He has written Fermat's 't Enigma so surely he is in contact with people that are both credible and serious.

3) He did have someone check his book but thought the story was better the way he told it. (This was common on the TV show

*Numb3rs *which never let a mathematical truth get in the way of a good story.)

I find this unlikely since a better way to say it is

*we'll never know if Fermat had a proof!*
One problem with such mistakes is that it destroys his credibility on other math things he writes of. That info about Dr. Who I got from the book, but I suspect its correct. And the stuff about the math that appears in the Simpsons is checkable and seems correct. I give one example: in one episode you see in the background

3987

^{12} + 4365

^{12}= 4472

^{12}
which is correct on a calculator with only 10 digits of precision. Cute! But I have stopped reading any of the math history in the book for fear that I will get incorrect notions in my head.

However, back to my original question: Was there a time when people thought Fermat really had a proof?Was there a time when people thought there was an easy proof? When did that change?