Fermat's Polygonal Number theorem vs FLT?
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From: spudnik on 6 May 2010 01:34 yeah, that's the spirit. the problem is, in developing a model of n-dimensional figurate numbers, that could be used to make the contradiction. (Conway and Guy had a nice, elementary "Book of Numbers" -- as I recall -- with lots of pictures for 3D figurates; I even made some "new results" using it .-) and I'm *really* sure that Pell's equation could by deployed as well, or in comjunction, if you thought about it long enough. > I mean, he worked on Pell's equation too, so why not an > inductive extension of that? Or anything else? It makes no sense. thus: the actual problem was in 1895, when Svente Ahrrhenius didn't bother to model an actual glass house at a particular lattitude ... and neither did anyone who had a computer in the climate lab. on the other hand, he probably didn't get the first Nobel for *that*, any way. thus: you mean, F"L"T is easy for the Sophie Germaine primes? thus: in contrast to Magadin's assertion, below, the reality is that n=4 is the only case that is truly special, which Fermat apparently didn't notice, when he wrote the marginal note. (may be, that's what blew him off, when I noted it in another item .-) Fermat apparently did not have to prove n=3, 5 etc., nor any other composite power (the "easy lemma" in all elementary treatments of numbertheory with F"L"T .-) thus: .... but, he did see one key (old) result, that Fermat's "last" theorem is the same, when applied to rational numbers, as pairs of coordinates on the unit circle (or the associated Fermat curves, for powers greater than two. well, it's quite trivial, as they say, but it is a good way to attempt the problem, a la Ribet, Frey etc. through to Wiles' Secret Attic Project. there's a really good expository book on the stuff around Wiles "proof," _Fearless Symmetry_. thus: since Fermat made no mistakes, at all, including in withdrawing his assertion about the Fermat primes (letter to Frenicle), all -- as I've posted in this item, plenty -- of the evidence suggests that the "miracle" was just a key to his ne'er-revealed method, and one of his very first proofs. (and, I wonder, if Gauss was attracted to the problem of constructbility, after reading of the primes.) --Light: A History! http://wlym.TAKEtheGOOGOLout.com
From: ThinkTank on 5 May 2010 23:18 > In article > <1853636941.69435.1272978833774.JavaMail.root(a)gallium. > mathforum.org>, > ThinkTank <ebiglari(a)gmail.com> wrote: > > > > On May 4, 10:16 pm, ThinkTank > <ebigl...(a)gmail.com> > > > wrote: > > > > If Fermat's original proof of the Fermat > Polygonal > > > Number > > > > Theorem has not yet been found, why are > > > mathematicians so > > > > quick to assume that Fermat did not have an > > > extension to > > > > that proof which may have led to the proof of > > > Fermat's Last > > > > Theorem? It seems to me the key to > understanding > > > FLT lies > > > > in a simple inductive proof for Fermat's > Polygonal > > > Number > > > > theorem. > > > > > > What nonsense. > > > > Yes, what nonsense? I barely said a thing. > > You said "It seems to me the key to understanding FLT > > lies in a simple induvtive proof for Fermat's > Polygonal > Number Theorem." I'll stand by my response. > And I stand by my response to your response. Would you like to do this again? > -- > Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for > email)
From: ThinkTank on 6 May 2010 00:00 > yeah, that's the spirit. the problem is, > in developing a model of n-dimensional figurate > numbers, > that could be used to make the contradiction. > (Conway and Guy had a nice, elementary "Book > of Numbers" -- as I recall -- with lots of pictures > for 3D figurates; I even made some "new results" > using it .-) > > and I'm *really* sure that Pell's equation could > by deployed as well, or in comjunction, if > you thought about it long enough. > Well, Fermat worked with Pell's equation but, as far as I know, he never claimed any results regarding it. Furthermore, I don't think that it has anything to do with Pell's equation, for the simple reason that Pell's equation is restricted to solely the power of 2, whereas the Polygonal Number theorem is a result regarding the addition of two polygonal numbers, of varying shape, to produce a third. It seems to me that the Polygonal Number theorem could be used to restate FLT as a sum of polygonal numbers, or figurate numbers in general, and thereby establish a recurrence. But this is mere speculation. However, I have developed a completely different proof, which does not rely on either of these, but rather on Galois fields, and an isomorphism between the integers and the reals. It is a complete proof, however, I may have made a mistake somewhere, so I am checking it. > > I mean, he worked on Pell's equation too, so why > not an > > inductive extension of that? Or anything else? It > makes no sense. > > thus: > the actual problem was in 1895, > when Svente Ahrrhenius didn't bother > to model an actual glass house > at a particular lattitude ... and > neither did anyone who had a computer > in the climate lab. > > on the other hand, > he probably didn't get the first Nobel > for *that*, any way. > > thus: > you mean, F"L"T is easy for the Sophie Germaine > primes? > > thus: > in contrast to Magadin's assertion, > below, the reality is that n=4 is the only case > that is truly special, which Fermat apparently > didn't notice, when he wrote the marginal note. > (may be, that's what blew him off, > when I noted it in another item .-) > > Fermat apparently did not have to prove n=3, 5 etc., > nor any other composite power (the "easy lemma" > in all elementary treatments of numbertheory > with F"L"T .-) > > thus: > ... but, he did see one key (old) result, > that Fermat's "last" theorem is the same, > when applied to rational numbers, > as pairs of coordinates on the unit circle (or > the associated Fermat curves, > for powers greater than two. well, > it's quite trivial, as they say, > but it is a good way to attempt the problem, > a la Ribet, Frey etc. through > to Wiles' Secret Attic Project. > > there's a really good expository book > on the stuff around Wiles "proof," > _Fearless Symmetry_. > > thus: > since Fermat made no mistakes, at all, > including in withdrawing his assertion > about the Fermat primes (letter to Frenicle), all > -- as I've posted in this item, plenty -- > of the evidence suggests that the "miracle" was just > a key to his ne'er-revealed method, and > one of his very first proofs. (and, > I wonder, if Gauss was attracted to the problem > of constructbility, after reading of the primes.) > > --Light: A History! > http://wlym.TAKEtheGOOGOLout.com >
From: Tim Little on 6 May 2010 08:19 On 2010-05-06, ThinkTank <ebiglari(a)gmail.com> wrote: > However, I have developed a completely different proof, which does > not rely on either of these, but rather on Galois fields, and an > isomorphism between the integers and the reals. I'd like to see this claimed isomorphism between the set of integers and the set of reals. In fact, I'd be extremely interested if you could produce even a mere bijection. - Tim
From: Gerry Myerson on 7 May 2010 01:44
In article <677236230.80457.1273132878888.JavaMail.root(a)gallium.mathforum.org>, ThinkTank <ebiglari(a)gmail.com> wrote: > ""...an isomorphism between the integers and the reals..." Thanks for removing any trace of doubt about your mathematical knowledge. -- Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email) |