Fermat's Last Theorem simple proof impossible?
in [Math]
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From: spudnik on 5 May 2010 16:49 the second root of one half is jsut the reciprocal of the second root of two -- often obfuscated as the second root of two, divided by two -- but the rest is indeed totally obscure or ridiculous. since Fermat made no mistakes, at all, including in withdrawing his assertion about the Fermat primes (letter to Frenicle), all -- as I've popsted 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. (I wonder, if Gauss was attracted to the problem of constructbility, after reading of the primes.) thus: so, you applied Coriolis' Force to General Relativity, and **** happened? > read more ยป --Light: A History! http://wlym.TAKEtheGOOGOLout.com
From: spudnik on 6 May 2010 00:47 .... 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 00:54 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. --Light: A History! http://wlym.TAKEtheGOOGOLout.com
From: Gerry Myerson on 6 May 2010 01:54 In article <1969189445.79703.1273112696073.JavaMail.root(a)gallium.mathforum.org>, DRMARJOHN <MJOHNMAR(a)AOL.COM> wrote: > To illustrate: the 10th root of .5 is .999323327...an irrational number. > .99932328 ends in an 8, above the 7... of the root of .5. .99932327 ends in a > 7, which is less than the 7... of the irrational root. This is what I call a > pair. Factoring each gives an An of .500000...ending in an even number (for > all 8s it is a 2 or 4 or 6 or 8). The Bn is .499999...ending in 1 or 3 or 7 > or 9, an edd number. Adding an even and an odd number can never equal the > even number 1.000. What does "factoring" mean to you? To the rest of us, it means writing a natural number as a product of two smaller natural numbers. I don't think it means that to you, since somehow it gives an An of .5 or something, which the kind of factoring the rest of us do doesn't do. It's a bad idea to try to do mathematics using words mathematician use but using them to mean things mathematicians don't mean. What does it mean for .50000... to end in an even number? Again, mathematicians use the dots to indicate that the expansion doesn't end at all. If what you mean is a finite decimal expansion that ends in an even digit, then say that. What does "for all 8s it is a 2 or 4 or 6 or 8" mean? I can't even begin to guess what you have in mind here. "Adding an even and an odd number can never equal the even number 1.000." Most of us think the number 1.000 is an odd number, but never mind; I suppose you mean that if you have two natural numbers and one is even and one is odd then their sum is odd. I don't know how it is in psychology, but in mathematics, we have developed a vocabulary that enables us to say what we mean, not sorta kinda almost what we mostly mean. > May I begin again? > For .5000 + .5000 = 1.000 (An + Bn = Cn = 1.000): > Consider the roots of .5000 from the exponent 3 to the exponent N. Why start with the exponent 3? What happens if you start with 2 instead? If everything you do for 3 works for 2, then anything you come up with for x^3 + y^3 = z^3 will also apply to x^2 + y^2 = z^2, which ought to tell you something. > As the > roots approach infinity, I guess you mean, as you take n-th roots with n approaching infinity. > A approaches 1.000. As it approaches infinity, > consider a pair of rational roots just above and just below the irrational > root of .5000. Call this An and Bn. At N-1 root, there are also a pair of > rational roots. At N-2, at N-3...descending to the third root, there are > pairs of rational A's and B's. This contains all the possible A's and B's. > Note that this approach does not consider all the domain of the third power, > instead it goes from the exponent 3 to the exponent N. The question, is this > all-possible-As-and-Bs a finite quantity or an infinite quanity? If you fix N, and take just one rational number either side of the n-th root of 1/2 for each n, then you're talking about a finite quantity of A_n and B_n. But if you let N go to infinity, or if for any particular n you look at all the pairs you can get by truncating the n-th root at different numbers of decimal places, then you're talking about inifnitely many A_n and B_n. -- Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: DRMARJOHN on 6 May 2010 02:56
Thank you for your comments. I understand about my naive vocabulary is different tnan yours. But you understand my last entry- I also meant that there are an infinite number by letting N go to infinity. I have two other questions, and my answers, that I think my vocabulary will convey what I mean: First,an answer to your question about N_2 vs N_3. There is a structural difference betwen the realm of FLT and that of square roots. The change between A and A_2 is in a straight line. The change between A and A_2 and A_3 is what can be called a curve. The difference between .9 and .9_2 and .9_3 is .09, then .081. If you graph .9_10 (.9 to the 10th power) and you connect the dots I daresay you will draw a curve. My question is, has anyone asked this naive question and seen the high school math answer? My second question: has anyone presented the naive observation that the realm of FLT is different than other equations, in that in FLT an odd number can only be multiplied by an odd number, an even number by an even number. If you graph .8_10 below .9_10 you will see that these curves diverge, that .8 changes faster than .9. If you imagine an irrational curve between these two. at the nth position, the lower curve will be further away. Do it this way: plot .7_10. At A_N. the distance between .7_N and .8_N is more than the distance between .8_10 and .9_N. In my mind I think of these as structural differences. It is these factors that lead me to see FLT the way I do. I also plot the hypothetical curves from A to A_N, rather than view all the possibilities for any one exponent. I would suggest that once Fermat presented a proof for the exponent 4, all followed him, and ran into problems. |