Fermat's Polygonal Number theorem vs FLT?
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From: Pubkeybreaker on 7 May 2010 07:29 On May 7, 7:20 am, "Achava Nakhash, the Loving Snake" <ach...(a)hotmail.com> wrote: > On May 6, 1:00 am, ThinkTank <ebigl...(a)gmail.com> wrote: > > > > yeah, that's the spirit. the problem is, > > > in developing a model of n-dimensional figurate > > > numbers, > It says nothing about one polygonal number being representable as a > sum of 2 polyngonal numbers. None of 3, 4, 5, 6, etc. are equal to 2, > and the representation is of ALL INTEGERS and not of another figurate > number. Again, the connection with Fermat's Last Theorem eludes me is > almost surely completely bogus. Note also that the Fermat equation is an algebraic plane CURVE, whereas in talking about polygonal numbers, the OP is now talking about varieties on SURFACES --> A totally different problem. And the OP is too ignorant to realize it.
From: ThinkTank on 7 May 2010 04:50 > On May 4, 5:16 am, 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. > > Why would this extend to Fermat's last theorem, when > it is about lots > of numbers of a certain type adding to any number, > whereas Fermat's > Last Theorem is about two numbers of ceratain types > adding to another > number of that same type? They are completely > different issues, and I > can't imagine how solving the one would shed any > light on solving the > other. I'm not sure how you can say they are a completely different issue. There are many similarities between the two problems. Most notably, both are statements about sums of n-symmetric shapes, where n>2. Stated mathematically: a^p + b^p = c^p => a + b == c (mod p) is the p-symmetric reduction. Furthermore, notice that Fermat's Last Theorem (FLT) is a statement about inequality, whereas the Fermat Polygonal Number Theorem (FPNT) is a statement about equality. So, it seems reasonable that the inequality of FLT may be stated as recurrence regarding the symmetry of hypercubes. Or, perhaps, it may be used to set a lower bound on the number of n-cubes needed to sum to an n-cube. However, it seems to me that a simple proof to the Polygonal Number Theorem must be quite spectacular, considering that it has not been found yet. > I mean, he worked on Pell's equation too, so > why not an > inductive extension of that? Or anything else? It > makes no sense. > > > Regards, > Achava
From: Pubkeybreaker on 7 May 2010 09:34 On May 7, 8:50 am, ThinkTank <ebigl...(a)gmail.com> wrote: > > On May 4, 5:16 am, ThinkTank <ebigl...(a)gmail.com> > > wrote: > I'm not sure how you can say they are a completely > different issue. There are many similarities between the > two problems. Most notably, both are statements about > sums of n-symmetric shapes, If you don't know the difference between a curve and a surface, I give up. You are so totally clueless about mathematics that you are not even aware of the DEPTH of your ignorance. However, in this particular instance, I will tell you the difference. Varieties over SURFACES have many more 'degrees of freedom' than varieties over curves. A^4 = B^4 + C^4 + D^4 *HAS* solutions. Infinitely many. They are different problems.
From: ThinkTank on 7 May 2010 06:11 > On May 7, 8:50 am, ThinkTank <ebigl...(a)gmail.com> > wrote: > > > On May 4, 5:16 am, ThinkTank <ebigl...(a)gmail.com> > > > wrote: > > I'm not sure how you can say they are a completely > > different issue. There are many similarities > between the > > two problems. Most notably, both are statements > about > > sums of n-symmetric shapes, > > If you don't know the difference between a curve and > a surface, I give > up. > You are so totally clueless about mathematics that > you > are not even aware of the DEPTH of your ignorance. > Do you NOT understand the concept of VOLUME and figurate numbers??? You are so totally clueless about mathematics that you are not even aware of the VOLUME of your ignorance. I will tell you the same thing I told Gerry. I go by the Golden Rule. If you want to throw around meaningless Ad Hominems, and waste my time, I will be MORE than happy to reciprocate. And, if you then continue, I will ignore you. Both you and Gerry need to GROW UP. I expect MUCH MORE from people who claim to be educated. Were you guys denied your tenure or something??? You've both got anger issues you need to deal with. > However, in this particular instance, I will tell you > the difference. > > Varieties over SURFACES have many more 'degrees of > freedom' than > varieties over curves. A^4 = B^4 + C^4 + D^4 > *HAS* solutions. > Infinitely many. > > They are different problems. >
From: ThinkTank on 7 May 2010 07:07
> On May 7, 6:26 am, ThinkTank <ebigl...(a)gmail.com> > wrote: > > > In article > > > > <677236230.80457.1273132878888.JavaMail.r...(a)gallium.m > > > athforum.org>, > > > ThinkTank <ebigl...(a)gmail.com> wrote: > > > > > > ""...an isomorphism between the integers and > the > > > reals..." > > > > > Thanks for removing any trace of doubt > > > about your mathematical knowledge. > > > > And you, Gerry, have proven that you do not READ > the > > posts. Besides that fact, I was speaking informally > to > > save time. Perhaps you have all the time in the > world to > > waste, but I do not. > > Gerry discerned the content of your posts quite > clearly. No, he did not, and apparently you don't READ either. He did not quote me correctly. > I stand with him. You are a crank, and you are > mathematically > clueless. Your claim of an You have no evidence to support that claim. Your quote is not even correct. You, Gerry and Tonio are grasping at straws here. I have made absolutely NO mathematical mistakes in ANY of my posts so far. NONE. Not ONE. Until you, Gerry or Tonio can point out a contradiction or mistake in my statements, you have no case. > > "> > > ""...an isomorphism between the integers and > the > > > reals..."" > > make this a certainty. |