My friend and I were discussing Fermat’s Last Theorem the other day, about how Fermat wrote in the margin of a book that “I have discoverd a truly remarkable proof of this thoerem which this margin is too small to contain”. My friend was suggesting perhaps Fermat actually meant this to be a joke…
Had anyone else thought the same way too?
Yes. IIRC, Fermat was known for winding people up. Either that, or he’d thought he’d discovered a proof when he really hadn’t.
Great gosh, then why on earth did people still spend so much time working on it, given Fermat’s reputation?
Because no matter whether Fermat had a proof or not, the problem was still a valid one.
I think Dominic Mulligan was joking. (If you’re not, Dominic Mulligan, please give a citation that he liked “winding people up.”) I don’t think that Pierre Fermat was known for making up fake theorems. What he was known for was coming up with conjectures or theorems which he claimed to have proved which were later confirmed to be true. Fermat’s Last Theorem is so called because it was the only one of the theorems/conjectures that he wrote down which wasn’t either proved or disproved within a few years after his death. But that’s not the basic reason that it became famous. It’s famous because it’s a mathematically interesting theorem.
The general supposition among mathematicians for the past one hundred years or so is that Fermat thought he had a proof, but he made a mistake. Someone has even come up a supposed proof of the theorem (which is consistent with the mathematics that Fermat knew) that appears to prove the theorem but which has a subtle mistake in it. What’s clear is that he couldn’t possibly have come up with Wiles’s proof of the theorem. That requires developing huge areas of mathematics that weren’t created until centuries after Fermat died.
Regarding Fermat as a practical joker, it seems that I was mistaken. I could have sworn that I had read it somewhere perhaps in the Simon Singh book?), but the best that I can do is offer this cite (see Nat Silver’s post) which states that it was his son who was a practical joker and all of Fermat’s papers had passed through him before being published.
In my opinion, the most likely possibility was Fermat was thinking about the theorem one day and thought he’d invented a proof. He wrote down his note to remind himself. But later, he worked it out and realized his proof didn’t work (there are several false proofs known). So to Fermat the note was an unimportant dead end and he never realized the number of people who would try to follow its lead. Keep in mind, Fermat lived another 27 years after writing his note, so he had ample opportunity to write out his complete proof if he had one.
My thoughts were that it was called “last theorem” because he died before he could write down the proof. It may be something to do with the word “last”… Why did they call it “Fermat’s Last Theorem” ?
If I understand correctly, Fermat never intended anyone else to see his marginal note. So if it was a joke, it would have been a very private one. (It would also explain why Fermat didn’t go back and correct his claim if he later realized that his “proof” was flawed. i.e. basically what Little Nemo said.)
Because they’re proved all the other ones, and this was the last one remaining unproven.
There’s also a relatively simple proof of FLT in the case that n = 3, and it’s possible that Fermat may have discovered that and believed that it would generalize.
It’s worth noting that standards of proof were not as rigorous in Fermat’s day as they are now. There are plenty of examples of papers from before the 20th century where mathematicians simply made huge leaps of logic or stated “this is obviously true”. Such behavior would shock modern mathematicians.
I can’t tell if some people in this thread are aware that Andrew Wiles solved Fermat’s Last Theorem several years ago. In any case, he believes that Fermat was either kidding or joking because the proof was very long and difficult.
I’ve read Singh’s book fairly recently. Fermat was well known for refusing to state proofs on his problems to rile other mathematicians, and did this on many occasions. But he was also an excellent mathematician and had actually worked out solutions to his other problems, even if he wouldn’t publish them. All hsi other theorems did have solutions. FLT was likely intended to annoy, but that doesn’t mean that Fermat hadn’t worked out at least a partial solution to the problem. If he did so, there is no reason at all to believe his solution was anything close to Wiles – I don’t think anyone believes this to be the case.
related question:
the theorem has been solved, but by a modern method using 20th century math.So this proof is definitely not the one that Fermat himself may have found.
Are mathematicians satisfied by this? Is anybody still interested in finding Fermat’s original proof, (if it existed?)
I don’t think any mathematicians exist who continue to think that Fermat had found a valid proof. Probably only amateurs are still looking for it.
It’s well known what the proof he was referring to was likely to be, and its inadequacies are equally well known.
Here is a solution for n=3. I was disappointed that Singh’s book did not include a discussion of this “simple” case – not that I could have proved this myself given a million years… but I would have appreciated the insight.
http://meta-religion.com/Mathematics/Articles/fermats_last_theorem.htm
Even if someone discovered a brilliantly simple proof using only concepts which were known in Fermat’s time, it would be impossible to prove that Fermat found the same one.
During the centuries between Fermat scribbling his note and Wiles publicizing his proof, there have been lots of proposed proofs, all of which turned out to contain some subtle (or not-so-subtle) error. I think most mathematicians nowadays assume that Fermat found one of these.
On the other hand the theorem itself certainly looks simple enough that it seems as if there should be a straightforward proof, not requiring hundreds of pages of 20-th century math. I guess that’s why it has attracted (and continues to attract) so many crackpots: it just seems like the kind of puzzle which an amateur with a decent mathematical intuition should be able to solve in an afternoon or so…
Look at Wendell Wagner’s post (post #5). Most mathematicians today believe that Fermat thought he had a proof of the theorem, but later realised that he had make a mistake. He then started working on proofs of special cases of the theorem. There is no way he could have written Wiles’s proof or anything similar.
That post alluded to a solution with a mistake still outstanding. I wanted to make it clear that the proof is done now.