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I've occasionally caught myself wondering whether there is a completely different proof that could have been discovered using 17th century mathematics. Or is it

I've occasionally caught myself wondering whether there is a completely different proof that could have been discovered using 17th century mathematics. Or is it the case that the maths used by Andrew Wiles would have been understandable to Fermat? Namely, I'm wondering if Fermat was actually lying and he just got lucky that a proof was eventually found.

tome 13y ago
Fermat definitely couldn't have understood Wiles' proof in terms of the mathematics of his time, since it relies on a lot of theory that wasn't available then. It's technically possible that there's a proof that only requires the mathematics of that period, but since no one has found it yet it seems unlikely.
raverbashing 13y ago
Not lying, but probably thought quickly of a proof that later was revealed to not be a correct one (not sure Fermat got to this point)
- tehwalrus 13y ago
  I agree; if he didn't write it down (and test it), it was probably a simple one that was subtly wrong.
- andrewingram 13y ago
  That's a good point, for some reason it didn't occur to me that he might simply have been wrong...
dagw 13y ago
I don't have a copy of the book in front of me, but I seem to recall Simon Singh's book talked about simple, but flawed attempted proof that was found in the 19th century, that might have been what Fermat found.
rst 13y ago
More likely Fermat was just mistaken. There have been numerous flawed proofs over the years --- most likely Fermat stumbled on one of those, realized the error (hence, never published), but never bothered to correct the note he'd left in the margins on of one his books.
ColinWright 13y ago
It is almost certain that Fermat did not have a proof. He did have (effectively) a proof for the case n=4, but it seems he worked for some time without success on a proof for n=3. It seems unlikely he would do so if he really did have a general proof. I'm told that if all rings of the form a+b.sqrt(-p) are unique factorization domains then there is a fairly simple proof[0], and it's plausible that Fermat momentarily forgot that this was not in fact the case. If he found the "proof" that uses this "fact" he may have got carried away and written his marginal note, then moments later done a face-palm, and not bothered to erase or correct that note. [0] I do not speak from personal knowledge, and this may have been distorted. I believe the essence to be correct.
- andrewingram 13y ago
  Were Mathematicians generally convinced that a proof did exist and it was just a matter of time? Or was there a not-insignificant belief that it could be false?
  - ColinWright 13y ago
    The mathematicians I know and who had expressed an opinion generally believed it was almost certain to be true, but that the techniques and machinery were not yet available to tackle the problem. There was genuine astonishment when Wiles produced his proof, but no real disbelief, as might be expected had people felt it not to be true. It should be noted that it had been proven true for a very wide collection of possible exponents. You can read about the piece-wise progress here: http://www-history.mcs.st-and.ac.uk/HistTopics/Fermat%27s_last_theorem.html http://www-history.mcs.st-and.ac.uk/HistTopics/Fermat%27s_la... As a piece of trivia, in the Horizon program[0] mentioned elsewhere there's a point where J.H.Conway[1] (inventor of the game of "Life[2]" (don't ever ask him about it!)) is being interviewed, and he uses the word "prime." The editors dubbed the word "number" over the top, feeling that the audience might be put off, and knowing that "number" was accurate enough for the purpose. I don't know if Conway was ever told - I'll try to remember to ask him next time I see him. [0] http://www.youtube.com/watch?v=7FnXgprKgSE http://www.youtube.com/watch?v=7FnXgprKgSE (roughly the 8 minute mark) [1] http://en.wikipedia.org/wiki/John_Horton_Conway http://en.wikipedia.org/wiki/John_Horton_Conway [2] http://en.wikipedia.org/wiki/Conway%27s_Game_of_Life http://en.wikipedia.org/wiki/Conway%27s_Game_of_Life

tome 13y ago

Fermat definitely couldn't have understood Wiles' proof in terms of the mathematics of his time, since it relies on a lot of theory that wasn't available then. It's technically possible that there's a proof that only requires the mathematics of that period, but since no one has found it yet it seems unlikely.

raverbashing 13y ago

Not lying, but probably thought quickly of a proof that later was revealed to not be a correct one (not sure Fermat got to this point)

tehwalrus 13y ago
I agree; if he didn't write it down (and test it), it was probably a simple one that was subtly wrong.
andrewingram 13y ago
That's a good point, for some reason it didn't occur to me that he might simply have been wrong...

dagw 13y ago

I don't have a copy of the book in front of me, but I seem to recall Simon Singh's book talked about simple, but flawed attempted proof that was found in the 19th century, that might have been what Fermat found.

rst 13y ago

More likely Fermat was just mistaken. There have been numerous flawed proofs over the years --- most likely Fermat stumbled on one of those, realized the error (hence, never published), but never bothered to correct the note he'd left in the margins on of one his books.

ColinWright 13y ago

It is almost certain that Fermat did not have a proof. He did have (effectively) a proof for the case n=4, but it seems he worked for some time without success on a proof for n=3. It seems unlikely he would do so if he really did have a general proof. I'm told that if all rings of the form a+b.sqrt(-p) are unique factorization domains then there is a fairly simple proof[0], and it's plausible that Fermat momentarily forgot that this was not in fact the case. If he found the "proof" that uses this "fact" he may have got carried away and written his marginal note, then moments later done a face-palm, and not bothered to erase or correct that note. [0] I do not speak from personal knowledge, and this may have been distorted. I believe the essence to be correct.

andrewingram 13y ago
Were Mathematicians generally convinced that a proof did exist and it was just a matter of time? Or was there a not-insignificant belief that it could be false?
- ColinWright 13y ago
  The mathematicians I know and who had expressed an opinion generally believed it was almost certain to be true, but that the techniques and machinery were not yet available to tackle the problem. There was genuine astonishment when Wiles produced his proof, but no real disbelief, as might be expected had people felt it not to be true. It should be noted that it had been proven true for a very wide collection of possible exponents. You can read about the piece-wise progress here: http://www-history.mcs.st-and.ac.uk/HistTopics/Fermat%27s_last_theorem.html http://www-history.mcs.st-and.ac.uk/HistTopics/Fermat%27s_la... As a piece of trivia, in the Horizon program[0] mentioned elsewhere there's a point where J.H.Conway[1] (inventor of the game of "Life[2]" (don't ever ask him about it!)) is being interviewed, and he uses the word "prime." The editors dubbed the word "number" over the top, feeling that the audience might be put off, and knowing that "number" was accurate enough for the purpose. I don't know if Conway was ever told - I'll try to remember to ask him next time I see him. [0] http://www.youtube.com/watch?v=7FnXgprKgSE http://www.youtube.com/watch?v=7FnXgprKgSE (roughly the 8 minute mark) [1] http://en.wikipedia.org/wiki/John_Horton_Conway http://en.wikipedia.org/wiki/John_Horton_Conway [2] http://en.wikipedia.org/wiki/Conway%27s_Game_of_Life http://en.wikipedia.org/wiki/Conway%27s_Game_of_Life