FLT
John Leonard
I am an amateur at mathematics. I am curious about Fermat's Last
Theorem, however. Since it has been proved, after about 350 years with
something like 200 pages of mathematics, do you think that Fermat had a
proof as he claimed? If he did then I suppose it could not have been the
proof that we have. And if that is the case, do you think that his original
proof will ever be found?
John Leonard
William Rex Marshall
news:m_L17.112$8i.7...@news.intnet.net...
The general consensus is that Fermat was just as mistaken as the thousands
and thousands of other amateurs who also claimed to have found a general
proof. See also Underwood Dudley's 1992 book "Mathematical Cranks".
MAppell917
>Theorem, however. Since it has been proved, after about 350 years with
>something like 200 pages of mathematics, do you think that Fermat had a proof
as he claimed?
Yes, I do think he had a proof and I am in the minority who think he did.
> If he did then I suppose it could not have been the proof that we have.
Correct, I do not in any way believe that he had the Wiles proof or anything
remotely related to elliptic curves and modular forms.
>And if that is the case, do you think that his original proof will ever be
found?
Yes and no. Since Fermat never provided a lot of detail, no one knows what he
knew. Also, I don't think any more evidence will be found of what he had but I
think another proof will be discovered and most will probably agree (those that
believe he had a proof) that it's the one he probably had.
Some mathematicians agree that a less modern proof without 20th century
mathematics and more algebraic proof of FLT is possible.
Mike
John Leonard
true that Fermat presented his theorem without actually having had a valid
proof? Are we to conclude that his theorem was a conjecture, and a lucky one
at that?
John Leonard
"William Rex Marshall" <williamre...@hotmail.com> wrote in message
news:3b47a204$1...@news.actrix.gen.nz...
William Rex Marshall
mathematician, only to later realise that his "proof" was invalid.
Certainly, he never published it, and we have no details of it. He did,
however, publish a proof for n=4, but unfortunately this was a simple one
which could have easily been discovered by others, and does not generalise.
Dudley points out in his book that the name "Fermat's Last Theorem" was
really a double misnomer, as it wasn't known for sure to be a theorem yet
(at the time when Dudley's book was published), and it wasn't Fermat's last
anything. "Fermat's Conjecture" would have been a better name for it.
"John Leonard" <John.L...@wwc.com> wrote in message
news:DVM17.115$8i.7...@news.intnet.net...
Nat Silver
>
> I am an amateur at mathematics. I am curious about Fermat's Last
> Theorem, however. Since it has been proved, after about 350 years with
> something like 200 pages of mathematics, do you think that Fermat had a
> proof as he claimed?
The consensus is no.
Fermat could have made a common error, concerning unique factorization
which doesn't show up until n = 37.
Another possible scenario concerns Fermat's son, who was a practical joker.
All of Fermat's manuscripts passed through his hands. I don't believe
Fermat's copy of Diophantus' work exists any more. I also think
Fermat's marginal note could have been created by his son.
>If he did then I suppose it could not have been the proof that we have.
Gauss and Euler were knew more and were better mathematicians than
Fermat and couldn't come up with an elementary proof.
>And if that is the case, do you think
>that his original proof will ever be found?
IMHO, Fermat had no proof.
KK
"Nat Silver" <mat...@worldnet.att.net> wrote:
> Another possible scenario concerns Fermat's son, who was a practical joker.
> All of Fermat's manuscripts passed through his hands. I don't believe
> Fermat's copy of Diophantus' work exists any more. I also think
> Fermat's marginal note could have been created by his son.
And, anyway, in the best case it was a small remark he wrote at this
margin when reading this book. He did not send it in letter to anyone,
as he usually did with his discoveries. In contrast, he did make the
proof for n=4 public. Thus it is very unlikely that he didn't find a
mistake in what he formerly believed to be a proof.
Thus, it is quite remarkable that the claimed statement was indeed
true...
KK
Chris Thompson
John Leonard <John.L...@wwc.com> wrote:
> Thank You for your reply. The reason that I ask however is this: Is it
>true that Fermat presented his theorem without actually having had a valid
>proof?
Fermat "presented" lots of [correct] results without proof in his
correspondence, and the extent to which he had valid proofs is often
in some doubt in these cases.
But FLT is not one of these: he never "presented" it at all. It occurs
in marginal notes he made in his copy of Diophantos that date from the
1630s, and were published only after his death, some 50 years later.
It's abundantly clear from what he did later "present" (the cases of
exponent 3 and 4) that he no longer believed he had a proof of the
general case.
> Are we to conclude that his theorem was a conjecture, and a lucky one
>at that?
We should conclude that he thought he had a proof when he wrote the
marginal note -- the language leaves no room for doubt on that ---
and that he later realised that he hadn't.
Chris Thompson
Email: cet1 [at] cam.ac.uk
Stephen Montgomery-Smith
>
> John Leonard wrote:
> >
> > I am an amateur at mathematics. I am curious about Fermat's Last
> > Theorem, however. Since it has been proved, after about 350 years with
> > something like 200 pages of mathematics, do you think that Fermat had a
> > proof as he claimed?
>
That is some practical joke. I wish that my jokes are as far reaching.
--
Stephen Montgomery-Smith
ste...@math.missouri.edu
http://www.math.missouri.edu/~stephen
Mehdi Tibouchi
> He did,
> however, publish a proof for n=4, but unfortunately this was a simple one
> which could have easily been discovered by others, and does not generalise.
That comment is a little unfair, imho. Fermat did invent the method of
infinite descent, and his application of it to the case n=4 is really
clever. I doubt any other mathematician before Euler could have come up
with the same proof. Besides, the method does generalise (to at least
n=3, Euler, and 7, Pépin), and is even central in the proof of Mordell's
theorem on elliptic curves.
--
M. TIBOUCHI <med...@club-internet.fr>
Murphy's Law:
Anything that can go wr
a.out: Segmentation violation -- Core dumped.
W. Dale Hall
>
> > I am an amateur at mathematics. I am curious about Fermat's Last
> >Theorem, however. Since it has been proved, after about 350 years with
> >something like 200 pages of mathematics, do you think that Fermat had a proof
> as he claimed?
>
> Yes, I do think he had a proof and I am in the minority who think he did.
>
... (stuff deleted) ...
>
> Mike
Do you have any basis (aside from a vague notion that simply-stated
things can be proven simply) for this belief? Do you have any expertise
that suggests that such a proof is possible?
Aside from a certain perennial that sprouts regularly on sci.math and
spouts all manner of FLT-related jibberish (together with great loads of
abuse for anyone who cares to question his correctness), and a small
trickle of once-or-twicers on the same topic, I haven't heard of
*anyone* with a serious involvement in mathematics who holds to the
belief that Fermat actually had a proof.
I'm curious and wonder who cares to disabuse us all of the notion that
Fermat-believers are like Leprechaun-hunters.
Dale.
Nico Benschop
As mentioned by others, his 'FLT' was probably a hunch or some clue
(to which he later did not return, likely realizing it was incomplete).
Maybe even something like http://arXiv.org/abs/math.HO/0103051
regarding the cubic roots of 1 mod p^2 (prime p=1 mod 6) and FLT case_1.
Which he could have easily discovered with the methods of that time,
around 1637 - just after his FST (Small Thm).
-- NB
Nico Benschop
The disbelief in a direct approach, via residues mod p^k, to FLT
(case_1)
stems from Hensel's lemma (on p-adics), which implies that any solution
of normalized equivalence: x^p + y^p == -1 mod p^k (any k > 1) ...[1]
is an extension of such solution mod p^2.
This is taken to mean (wrongly;-) that no integer inequality can be
derived from the nec.condition [1], since k can be ANY positive integer,
also for limit k --> inf. ('asymptotic' p-adic solution).
Given any residue solution [1] for some k>1 (a necessary condition for
an integer solution), no p-th power integer solution, with terms <
p^{kp} can be derived from it.
In fact, taking such k-digit solution terms as integers < p^k,
their p-th powers imply necessarily inequivalence mod p^{3k+1},
thus at "triple precision" (for any precision k > 1).
This is a bootstrap type of phenomenon, 'breaking the Hensel lift';-)
The cubic roots of 1 mod p^k (p=1 mod 6) play a critical role here
[notice: just as the cubic degree of elliptic curves]:
a^3 == 1, with a + 1/a = -1 (mod p^k)
and its generalization to the 'triplet' rootform of FLT(case_1) mod p^k:
a + 1/b == b + 1/c == c + 1/a == -1, with abc == 1 (mod p^k).
Which btw is the general structure for ALL units in the unit-group
(not only p-th power residues mod p^k)
-- NB -- http://home.iae.nl/users/benschop/scimat98.htm
http://arXiv.org/abs/math.GM/0103014
Robin Chapman
>
>The disbelief in a direct approach, via residues mod p^k, to FLT
>(case_1)
>stems from Hensel's lemma (on p-adics), which implies that any solution
>of normalized equivalence: x^p + y^p == -1 mod p^k (any k > 1) ...[1]
>is an extension of such solution mod p^2.
The disbelief has nothing to do with Hensel's lemma, rather with the
fact that x^p + y^p = z^p (mod p^k) has nontrivial solutions
for all p and k.
>This is taken to mean (wrongly;-) that no integer inequality can be
>derived from the nec.condition [1], since k can be ANY positive integer,
>also for limit k --> inf. ('asymptotic' p-adic solution).
>
>Given any residue solution [1] for some k>1 (a necessary condition for
>an integer solution), no p-th power integer solution, with terms <
>p^{kp} can be derived from it.
Have you any *proof* for this?
>In fact, taking such k-digit solution terms as integers < p^k,
>their p-th powers imply necessarily inequivalence mod p^{3k+1},
Proof again?
> thus at "triple precision" (for any precision k > 1).
>This is a bootstrap type of phenomenon, 'breaking the Hensel lift';-)
>
>The cubic roots of 1 mod p^k (p=1 mod 6) play a critical role here
> [notice: just as the cubic degree of elliptic curves]:
> a^3 == 1, with a + 1/a = -1 (mod p^k)
>and its generalization to the 'triplet' rootform of FLT(case_1) mod p^k:
> a + 1/b == b + 1/c == c + 1/a == -1, with abc == 1 (mod p^k).
>Which btw is the general structure for ALL units in the unit-group
>(not only p-th power residues mod p^k)
>
>-- NB -- http://home.iae.nl/users/benschop/scimat98.htm
> http://arXiv.org/abs/math.GM/0103014
This is a reference to an MS claiming to give a proof of Fermat's
Last Theorem, but not actually containing a proof of Fermat's
Last Theorem.
------------------------------------------------------------
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
"His mind has been corrupted by colours, sounds and shapes."
The League of Gentlemen