## Monday, September 21, 2015

### When did Mathematicians realize that Fermat did not have a proof of FLT?

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

398712 + 436512= 447212

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?

1. My guess is that for a while they thought he had a proof and so they worked to figure out what it was knowing what they knew about his other work of the time and came upon an attempt or two that had subtle errors and figured out that he thought he had a proof but then when he went to formalize it realized he didn't and he moved on with his life. We it seems did not.

2. I don't know about the history after Fermat (if anyone knows, Singh does). However, on Fermat himself, the consensus is that Fermat believed he had a proof, but later discovered it had a flaw in it, possibly when it came time to actually work out the details.

It's interesting to speculate what "proof" Fermat thought he had.

3. not an expert in this area but my understanding is that there is some rough proof that "elementary" techniques cannot succeed related to the complex mathematical framework built up over last few decades (would like to know more details on that myself). ie a sort of impossibility proof. but interpreting those can be very tricky as RJLipton has argued in his blog.

4. Hi Bill,

I think that Singh does not make any statement to this effect in Fermat's Enigma. If anything, he in fact reiterates (or at least that's what I recall) what we all feel -- Fermat thought that he had a proof, but which must have been incorrect.

I would feel that with this Simpson's thing really your option 3 should be the explanation. This is most likely (and perhaps unfortunately) done with the intention of making the story "better".

5. This used to be a FAQ in sci.math something I used to maintain in a previous life:

https://cs.uwaterloo.ca/~alopez-o/math-faq/node26.html#SECTION00414000000000000000

6. I recall my abstract algebra professor in undergraduate giving me the modern version of a *specific false proof* of FLT using some basic algebraic number theory (trying to solve the equations over various rings like Z[\sqrt{-3}] and whatnot). He said that experts agree this is probably the technique that Fermat thought he had found for FLT, but because he did not write it down and check it thoroughly enough he did not see his mistake (that such and such elements used are not prime in whatever ring). I cannot remember the specific details.

7. In 1989 FLT was still open.

8. (This is Bill Gasarch)
The episoe was Shown in 1989 but took place in the far future. The writers didn't
know that FLT would be solved soon, so I don't really call that an error on their parts. After it aired I asked Larry washington, an number theorist when FLT would be solved and he said in the next 10 years'' which I call a very good estimate.
I also asked him about Rieman: not in my lifetime' and