Fermat's Last Theorem approach?
in [Math]
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From: ThinkTank on 5 May 2010 20:44 Suppose Fermat's Last Theorem could be proven for all n and a,b,c of the form: sum(i=0,k) d_i*2^(m*i), for all k>=0, and one m=n^3, where d_i is an element of {0,1}, and q_n is an integer constant for each n. Or, stated using regular expressions to describe the base 2 representation: a,b,c elements of {((0|1)0{m1})*(0|1)} where, m1=n^3 - 1. Does anyone know of a way to extend such a proof to all integer a,b,c? My intuition tell me no, because I am fairly certain have already found a very simple proof of FLT for those type of integers. Therefore, if it could be extended to all integers, the proof for FLT would be very simple. And, considering the amount of effort mathematicians have put into solving FLT, I doubt such a simple proof would be correct. But, it doesn't seem like a very large leap to go from a,b,c of the form described above, to all integer a,b,c. So, it seems reasonable to ask the question, does anyone know of whether such an extension is possible?
From: spudnik on 6 May 2010 00:58 you mean, it's 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. 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: Gerry Myerson on 6 May 2010 01:23 In article <1595104331.80090.1273121129807.JavaMail.root(a)gallium.mathforum.org>, ThinkTank <ebiglari(a)gmail.com> wrote: > Suppose Fermat's Last Theorem could be proven for all n and a,b,c of the > form: > > sum(i=0,k) d_i*2^(m*i), for all k>=0, and one m=n^3, > > where d_i is an element of {0,1}, and q_n is an integer constant for each n. What? "q_n is an integer constant for each n" but there's no other mention of q_n. "one m = n^3" but there is only one m. Maybe m*i was supposed to be m_i? -- Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: ThinkTank on 5 May 2010 21:27 > you mean, it's easy for the Sophie Germaine primes? > If the extension from a,b,c of the form described to all integer a,b,c can be made. Do you know of such an extension? > 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. > > 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: spudnik on 6 May 2010 01:37
"What the ****?" was my initial response, two, especially when he went into geek mode on "regular expressions." |