
| NAME | Jess P | |
| jessp54@hushmail.com | ||
COMMENTS |
Excellent web site, Thanks
Jess | |
| DATE | Saturday, February 6th 2010 - 12:55:22 PM | |
| NAME | BJ Edwards CPFA | |
| brian@b1je.orangehome.co.uk | ||
COMMENTS |
In my previous comments, the underlining of exponents (powers) have vanished, but can be identified as n or any figure after a letter symbol. R n has appeared as rn. In paragraph 2, line 11, 171 should read as 28.5. | |
| DATE | Friday, January 29th 2010 - 08:23:55 AM | |
| NAME | BJ Edwards CPFA | |
| brian@b1je.orangehome.co.uk | ||
COMMENTS |
Briefly, you note the exclusively Pythagorean (P) x2+y2 = z2 and r2 = 2(z-x)(z-y) and Fermat (F) of xn+2 + yn+2 = zn+2; but you rely in figures 4-10 on the Universal (U) equations and their components, which will accept any values you care to put into them, including all those from (P) and (F), of x+y – r = z, and therefore r =x+y-z, x = r+(z–y), y = r+(z–x), and z = r+(z–x)+(z–y). They are actually all the same equation. Your last Paragraph, Page 14, summarises this major reliance. Surprisingly, there is no simple, meaningful, linear equational relationship or connection between (P) and (F)3 or between any (F)n - all (F)rn equations are dedicated and unique, e.g. (F)r3 = 3(z-x)(z-y)(x+y). Try creating (P)r2, (F)r3 or any (F)rn from the (U)equations, or from each other. (F)r3 is not (P)r2r, as you use it, and (F)rn is not (P)rn in the extended model in the last Paragraph, Page 14. Detritus (D) is the e.g.1026 surplus created by cubing the e.g. (P) triple of 8,15,17 used in figures 1-3. (F)nxyzr’s are all different and (D)s would be infinitely variable in size, and shape. Try creating (P) in a triangle, by counting and folding, using z=19 or 22, or any other (F) infinite potential value which is not (P). Should this be found to be possible it might help to clarify the geometric and equational value of this single (P), schematic, triangular Euclidian progression.
Does Page 14, penultimate paragraph, refer to the (P) cubed start, giving the excess in an overall (P) cube that could be expected? The word “relationship” needs clarification when explaining the physical integer contents of a three dimensional shape. In Page 14, Paragraph 5 and the 1st Paragraph 3, the “r” triangles of Figures 8 and 9 may have triangular sides as in Figures 1-3, and “be a model for squared numbers” and “ generate (P) (square) triplets”, but this does not mean they “contain” them. The “r” triangles have an overall length shown of (P)r, and do not show geometrical numerical values because (P)r can be any length – are you saying a (P)z2 is in each digit of (P)r length with a triangular end? E.g.1026 divided by (P)r2 (36)=171; and it is necessary to show z, and x/y cannot be 17, and 8 or 15 in (F)n. So, essentially, as the cubic shapes in Figures 5-9 have been given no values or mathematical context, what are the (F) cubic equations for the (z-x), (z-y) shapes, and for the “z” and r2 “triangles”? You need to reconsider if you have proved FLT. There has, of course, always been a primitive (not “trivial”) (P) triple generator in the Babylonian equations. I am re-publishing my results, now partly to resolve apparent, and major, widespread misunderstandings. | |
| DATE | Wednesday, January 27th 2010 - 05:58:28 PM | |
| NAME | Sotirakis Peklivanas | |
| jim.peklivanas@hotmail.co.uk | ||
COMMENTS |
Thank you for publishing this proof. It was most enjoyable to read. I actually have a proof which is less than six lines long. a, b and c are not what they seem to be. But I can tell you that b is the variable that drives the other two. Symmetry is what allows n=2 to work.If you increase the size of a square you will find the 1+3+5+7+9+... lends itself to 1^2,2^2,3^2,4^2,.....where 1+3=4,1+3+5=9,...
This does not happen with n>2 as the inner shells of the higher dimensions fall short of any summation for a^n+b^n<>c^n. I do have the proof in algebraic form, as well as the infinite series of pythagorian triplets which satisfies 2n-1 and 2n. I WOULD BE INTERESTED IN ANY OTHER IDEAS THAT YOU HAD LEADING UP TO YOUR FINAL PROOF as I do not believe that John Wiles has actually provided a proof with integrity, but more a proof checked by friends in the know. The proof lies in simplicity and not complexity. my regards Jim P. | |
| DATE | Tuesday, January 12th 2010 - 12:10:27 PM | |
| NAME | BJ Edwards CPFA | |
| brian@b1je.orangehome.co.uk | ||
COMMENTS |
I congratulate you on what I hope is the simple solution, but will send information about my earlier publication, registration, and correspondence in 2000, in which I set out the full method of generating Pythag. triples from all even numbers, with implications and examples. | |
| DATE | Tuesday, December 29th 2009 - 07:24:33 PM | |
| NAME | Kris Walther | |
COMMENTS |
If you make a diagonal in a square, let's say 7 x 7, on a squared paper, and then in one of the triangles fill out the hole squares with twos, and the half squares with ones, then the sum will become 49. No odious in that. Toms explanation for Pythagoras is there by no miracle!
Because the twos represent a area double as the ones, Ballards conclusion that these have to be fill in with same numbers as cut of, is not true. A 'two' can easily be changed to 2 'ones', and reversed. Sorry for that. It was rather such a nice solution, you have maked. And sorry for removing your believes. | |
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| sally@juno.com | ||
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| DATE | Tuesday, September 22nd 2009 - 05:25:31 PM | |
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| DATE | Sunday, September 13th 2009 - 01:46:11 PM | |
| NAME | Harold | |
| h.merlin55@gmail.com | ||
COMMENTS |
A sad proof. Very good presentation for deriving x^2 + y^2 = z^2 triplets! | |
| DATE | Monday, August 31st 2009 - 09:29:30 AM | |
| NAME | Romeo155 | |
| romeo.115jr@rocketmail.com | ||
COMMENTS |
Nice presentation but a fallacy.
You have assumed that r is fixed - which should not be the case. This proof is simply unacceptable. | |
| DATE | Friday, August 28th 2009 - 09:29:27 AM | |
| NAME | ozirizis | |
| ozirizs@hotmail.com | ||
COMMENTS |
Good idea | |
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| DATE | Saturday, August 1st 2009 - 04:01:48 PM | |
| NAME | Arnold Madrid | |
| arnold.madrid17@rocketmail.com | ||
COMMENTS |
The proof is a fallacy. Ea is right when she says that you have "fixed" r^2 to equal 2(z-y)(z-x)! Changing the value of x,y or z won't not always yield r^2 = 2(z-y)(z-x) just as there are infinitely many obtuse triangles with integral sides!!! | |
| DATE | Tuesday, July 28th 2009 - 08:09:14 AM | |
| NAME | Ea | |
| Christy_Milan22@ymail.com | ||
COMMENTS |
You have "forced" r^2 to equal 2(z-y)(z-x)! This is a fallacy since x,y,z cannot be arbitrary. At least once of them is a variable. It x = r + (z-y), y = r + (z-x)and z = r + (z-x) + (z-y) then forcing r^2 = 2(z-x)(z-y) would automatically yield x^2 + y^2 = z^2!!! | |
| DATE | Monday, July 20th 2009 - 08:15:44 AM | |
| NAME | Megalith '82 | |
| megalith.82@ymail.com | ||
COMMENTS |
The construction of the model for x˛ + y˛ = z˛ and xⁿ + yⁿ = zⁿ very is impressive and in fact correct! However, you can generate x,y,z triplets by simply stacking "z" triangles without having to further manipulate the wedge geometrically. It appears that the "detritus" thing the proof mentions is redundant and only makes the proof absurd. | |
| DATE | Thursday, July 16th 2009 - 10:39:00 PM | |
| NAME | J.Donald | |
| james.donald@yahoo.com | ||
COMMENTS |
Unfortunately, this is a false proof.
Your generalized model assumes that the "wedge" which expands the z^2 triangle is the ONLY means to construct z^3 or z to any higher power. However, there are many methods by which one can construct z^3. Constructing a cube for a model comes immediately to the mind. (We do call ^3 "cubed" for a reason). The higher the power (n), the greater the number of models which may be constructed. The wedge is not the only means nor model. Since you forced z^n to be a wedge based on the z^2 triangle, you are assuming that x^2 + y^2 = z^2 prior to showing that any solution for x^n + y^n = z^n must reduce to a valid solution for x^2 + y^2 = z^2. In logic, this is begging the question or assuming the conclusion. Instead, you have proven a weaker case. Namely, if x, y, and z are nontrivial, integer solutions to x^2 + y^2 = z^2, then they cannot also be solutions to x^n + y^n = z^n unless n = 2. | |
| DATE | Monday, July 13th 2009 - 08:31:59 AM | |
| NAME | Cynthia Large | |
| cLargeZoo@yahoo.com | ||
COMMENTS |
Most impresssive! | |
| DATE | Saturday, June 27th 2009 - 02:00:09 PM | |
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| tucker@telusplanet.net | ||
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| DATE | Wednesday, June 17th 2009 - 07:26:15 PM | |
| NAME | manjunath bs | |
| abc123.m@gmail.com | ||
COMMENTS |
excellent..! | |
| DATE | Tuesday, January 6th 2009 - 10:20:24 AM | |
| NAME | Parviz Moarefi | |
| pmoarefi@bellsouth.net | ||
COMMENTS |
Simple and to the point. Well done. | |
| DATE | Thursday, December 4th 2008 - 08:48:44 AM | |
| NAME | Oscar Lindo | |
| judsoss@aol.com | ||
COMMENTS |
The work presented is excellent. The conclusion that X^n + Y^n = Z^n must fit into a pythaforean construct is an absolute truth.
I am amazed that this work is not part of main steam math. | |
| DATE | Tuesday, December 2nd 2008 - 07:39:21 AM | |
| NAME | t |
| DATE | Saturday, November 29th 2008 - 04:17:13 PM |
| NAME | inba jeyanthi | |
| inba_jey@yahoo.com | ||
COMMENTS |
i want to read the book | |
| DATE | Tuesday, November 11th 2008 - 12:33:14 AM | |
| NAME | Jakub Tatomirj | |
| kubana44@yahoo.co.uk | ||
COMMENTS |
Greatings to all of yee | |
| DATE | Saturday, October 25th 2008 - 06:03:06 AM | |
| NAME | Julita Santos | |
| talofofo53@yahoo.com | ||
COMMENTS |
Thank you for making it available online. | |
| DATE | Monday, September 29th 2008 - 09:32:28 AM | |
| NAME | Tias | |
COMMENTS |
I dont understand | |
| DATE | Friday, September 26th 2008 - 07:24:18 AM | |
| NAME | Alan Bunyan | |
| alanbunyan@yahoo.co.uk | ||
COMMENTS |
Unlike most of the guests here, I'm not a mathematician (not even an amateur one!), just a layman fascinated by this centuries-old debate. I was simply struck, on reading your fascinating article, by the following question: could there be any inherent/fundamental connection between any/all of the following basic mathematical facts? :-
(1) there are only TWO types of integers that can exist, even and odd; (2) when any TWO numbers of the same type (be that odd or even) are added together, the result will always be even, and only when TWO numbers of different types are added together will the result be odd; (3) when either any TWO even numbers or any TWO numbers of different types are multiplied together, the result will always be even, and only when TWO odd numbers are multiplied together will the result be odd; (4) the Pythagorean equation (so far as anyone can tell) does not work with any genuinely multiplicative power other than TWO (i.e. excluding 0 and 1). There just seems to be something ineffably primal about all those 2's!! (Apologies if, as I suspect, I've done no more than state the obvious!) | |
| DATE | Thursday, September 4th 2008 - 01:25:29 AM | |
| NAME | J. IVAN NATALARAY | |
| JIVANNATALARAY@GMAIL.COM | ||
COMMENTS |
no comment | |
| DATE | Saturday, August 30th 2008 - 11:04:04 PM | |
| NAME | J. IVAN NATALARAY | |
| JIVANNATALARAY@GMAIL.COM | ||
| DATE | Saturday, August 30th 2008 - 11:02:42 PM | |
| NAME | Greg | |
| wyzgreg@rogers.com | ||
| DATE | Monday, June 23rd 2008 - 07:57:54 AM | |
| NAME | FlashDiaz | |
| FlashDiaz@hotmail.com | ||
COMMENTS |
Great site. I was particularly intrigued by your visual representations of squares as triangular sums of numbers. I wonder if a better n-dimensional model can be constructed. In 3-space, one can create a triangle consisting of a 'base' row of ones, followed by increasing multiples of 6. The model remains 2-dimensional, but describes cubes. Finding Fermat triplets follows your layout for squares. | |
| DATE | Thursday, June 12th 2008 - 12:56:35 AM | |
| NAME | Mick |
| DATE | Sunday, May 25th 2008 - 09:37:08 PM |
| NAME | ŔĚŔçŔ˛ | |
| leejaeyul5@yahoo.co.kr | ||
| URL | http://blog.empas.com/leejaeyul5/ | |
COMMENTS |
Our proofs are perfect and plain.
4 Color Theorem proof of the regions on global surface Two each methods about FLT proof All Pythagorean triples cannot be the power numbers. We believe in the Fermat. And we believe that the space and the matters come into existence, when the numbers come into existence and we also believe that all cosmic materials and lives change but the number theory cannot change now and forever. Thanks. | |
| DATE | Thursday, May 15th 2008 - 03:33:12 PM | |
| NAME | sama | |
| sanam718@yahoo.com | ||
COMMENTS |
a history of time from black holes | |
| DATE | Wednesday, May 14th 2008 - 11:41:22 AM | |
| NAME | Berrier Dale Byrom | |
| dbyrom51@plateautel.net | ||
| DATE | Saturday, May 3rd 2008 - 08:58:51 AM | |
| NAME | Karsten Tjugen | |
| kars-tj@online.no | ||
COMMENTS |
Working at the moment with an idea for a SHORT proof for last FERMAT assumption,I find you approach interesting.
I find also your proof to be fairly long, and I am looking for a shorter one. | |
| DATE | Thursday, December 6th 2007 - 05:39:09 AM | |
| NAME | Mario De Paz | |
| mario.depaz@fastwebnet.it | ||
COMMENTS |
I am with you that Fermat had a simple proof of his last theorem. At present I cannot say that your proof is correct for any n: I have not studied it enough to understand its deep significate.
I appreciate much your efforts which deserve attention. | |
| DATE | Monday, October 29th 2007 - 12:06:38 AM | |
| NAME | Sophie Germain | |
COMMENTS |
Nice proof for n=3. I will be very pleased, if you could show the same for n=7. I think, you will run into trouble! | |
| DATE | Wednesday, April 18th 2007 - 11:00:34 AM | |
| NAME | Walther Bush | |
COMMENTS |
Trinare system: 1^3=1, 2^3=22, 10^3=1000, 11^3=2101, 12^3=11122, 20^3=22000, 21^3=110201, 22^3=200222, 100^3=1000000, 101^3=1101001, 102^3=1211022, 110^3=2101000, 111^3=10000101.
Do you see the pattern? | |
| DATE | Friday, February 16th 2007 - 01:32:27 PM | |
| NAME | MPENDULO V. MAGAGULA | |
| 126941@uniswacc.uniswa.sz | ||
COMMENTS |
no comments | |
| DATE | Wednesday, January 10th 2007 - 06:01:19 AM | |
| NAME | Jo | |
COMMENTS |
This query was raised in the 1600's??? Since the mid 70's(that's 1970's), my favourite professor at Queen's University spent the greater part of his life wrestling like Jacob with this one...Out of sincere compassion and love for this gentleman and others that have been preoccupied with this one, may I respectfully suggest reading Phil 4:8 then get some rest! Blessings in Him. | |
| DATE | Monday, January 8th 2007 - 05:49:35 PM | |
| NAME | Clefort | |
| ipub@charter.net | ||
COMMENTS |
Thanks for sharing your proof and your invaluable notes.
CL. | |
| DATE | Monday, December 4th 2006 - 04:04:08 PM | |
| NAME | ajumunisha | |
| ajumunisha_begam | ||
COMMENTS |
i want the real proof of fermats last theorem | |
| DATE | Sunday, November 26th 2006 - 10:17:25 PM | |
| NAME | Allan H H | |
COMMENTS |
x = a – b, y = a + b <=> (a – b)^3 + (a + b)^3 = z^3 <=> z^3 = 2a(a^2 + 3b^3)
x = c - d, y = c + d <=> (c – d)^3 + y^3 = (c + d)^3 <=> y^3 = 2d(d^2 + 3c^3) x = a – b, y = a + b <=> x^3 + (e – f)^3 = (e + f)^3 <=> x^3 = 2f(f^2 + 3e^3) 2a = (x + y), 2d = (z – x), 2f = (z – y) <=> x = (2a – 2d + 2f) / 2, y = (2a + 2d – 2f) / 2, z = (2a + 2d + 2f) / 2 (a – d + f)^3 + (a + d – f)^3 = (a + d + f)^3 <=> a3 – d3 – f3 – 3a2d + 3ad2 – 3a2f + 3af2 – 3d2f – 3df2 – 18adf = 0 (a – d – f)^3 <=> a3 – d3 – f3 – 3a2d + 3ad2 – 3a2f + 3af2 – 3d2f – 3df2 + 6adf (a – d – f)^3 = 3(2a)(2d)(2f) This is absurd, but it is only for proof n=3. Like your story, higher n is more complex. You can’t proove all n, just telling about n=3. | |
| DATE | Saturday, October 21st 2006 - 02:32:32 PM | |
| NAME | E. Terrell | |
| eterrell00@netzero.net | ||
COMMENTS |
Well... What can I say...? ...your proof is interesting - But! Like Andrew Wiles... Its wrong! Still, if you are indeed interested in the proof and the meaning of the proof of Fermat's Last Theorem - read:
http://www.ietf.org/internet-drafts/draft-terrell-math-quant-ternary-logic-of-binary-sys-01.pdf In other words, the proof of Fermat's Last Theorem / Pythagoras Theorem is established by the Distributive Law... The Alternate Mathematical it represents - The Binary System...! | |
| DATE | Tuesday, June 6th 2006 - 11:50:01 PM | |
| NAME | james w. barclay | |
| rom1210@verizon.net | ||
COMMENTS |
Here goes: Fermat was a man of the 17th century. A man who lived in the middle of the greatest creative era of the renaissance. Especially in building and architecture. He was well-educated, was a thinker, a lawyer, a killer Bridge player and an amateur mathematician. He enjoyed the whole thing that was going on. One can imagine him, several days out on walks or coming home from court or his office and seeing all this fantastic stuff. One day he looks up and notices the stonemason's crane, not changed in hundreds or even a thousand years. Not even DaVinci could improve it.
It was a right triangle of thick wood so steady it could, and did, haul stones, mortar, other materials machine and equipment up to where the masons were working. Could he improve up this? He may have thought: You know, I could make some money improving of this concept which is ancient." He knew, as well as any person who had studied geometry or trigonometry, that for x(n)+y(n)=Z(n) and he would certainly have been taught that n=0,1or 2, but that any hgher number of n would not work. It was an absurdity and invalid. He goes back home and starts playing with the idea of making a new crane. But everytime he comes to stability factor, every design fails. He must, of course, make a crane of the same simplicity, but better. Is he nuts? No. He just thinks he can come oup with something better. Its a game of sorts. But, should he indeed bump into another unique solution, he might have something that will make him famous or at least thought of as a very clever boy. And he knows he's not the smartest guy in Paris. He fails and, in doing so, one day simply puts down that the solution to the Pythagoran statement of the Right Triangle has never been solved to be wrong, and never will. And what does he use for his proof? He simply defies anyone to prove him wrong. Its unique because he puts in mathematical form so that it must be. He's not worried about how many variations or new algebras or diagrams. Why put them in? They don't work. Unfortunately, he puts down that he can't fit the fact that it is unique in the margins of his papers. He dies. Fermat is known as a lawyer and killer card player, and a lawyer and judge.His papers, or some them get collected. Of course, he has been famous. Do, obviously someone wants possession of his papers, just for the art of collecting. Nothing is unusual or really out of the ordinary. After all, who gave a box of wig powder about Sir Christopher Wren's assistant? Who the dickens cared about King Louis' accountant, Montesque? Back then, civil fame was a very short game. Folks just didn't have much of a lifespan. and the average height of a man was like 5'4" or less. The normal height of a house's ceiling was about 6'. Mathematicians got ahold of it and said: "Here is a great genius. HE must have known so much about mathematics. He was smarter than us. I know, we'll use all our higher math to solve this. It got famous, too. A head finally came when a little English mathematical whizz kid found the theorem in a public library and wanted a go at it, an obsession it became and it dominated his life. Its all he really wanted to do: solve that claim of uniqueness for the Pythagoran Theoram. Finally, when he had the means to do it, he spent virtually all his time on it using the most sophisticated mathematics in the known world and molding them into what he formulated to be a solution. In other words he want to prove "uniqueness" anther way of proving a set of negatives. Pretty wild. One day he looked and said: "This is it." He presented it to a very elite group of pure mathematicians and they went nuts, cheering and hooting and so on. He goes back a hero. His paper goes to the dreaded Peer Review. It comes back, "close, but no cigar. Therefore, you didn't do it." He goes back home and works on the thing for about another six months and he's getting nowhere- and worse, everyday and night. One morning he give it a break and goes out for a walk in the woods. Couple of hours I guess. When he comes back, his assistant has made some notes. HE uses the notes and straightens out his math. He galnces at it and then looks. This guy, Andrew Wiles, literally breaks down and sobs. He cannot even look anyone straight in the face and talk about it without bursting into tears. Needless to say it is published and he's again the hero of pure math. Its published, he's on the news, he's a superstar with everyone, but the public and other scientists outside his little sphere. finally, there is this guy, Simon Singh, a mathematician and famous writier who has already had a best seller on cipers. He writes "Fermat's Last Theroem" abouth this journey, excluding probably how Fermat actually did it. WGBH-TVs "Nova" does an episode according to Singh and then he really is famous. There is only one thing. The kid looks so fragile he might break in half. Singh kind of goads him to tell the story about how he know he actually had it. Again, Wiles breaks down in tears and really can't finish. Its heart wrenching to watch. But, not only had this little guy Wiles solved Fermat, he also solved the mystery of Elliptical Curves and a thing called the "Tanayama-Simura Theory" which stated that Tanayama-Shimura math is exactly equal to Elliptical Curves. Now it becomes the "Tanayama-Shimura Principle". He also proves a couple of theorists wrong. (Tanayama committed suicide in 1959 due to his perception of his failure in his life long quest.) When Shimura was asked about the prinicple he said: "They should have listened to us. It would have save them a lot of work." But no one, no one ever thought to examine the history around Fermat and his intrigueing study of the stonemason's crane, which is still used today. Sometimes, oftimes, the answere lies in the most obvious place and under the most common circumstances. Back in the old days of science they had to. Modern science forgot it and the old pronouncement of Richard Feynman echoes: "If you can't explain you theory to me in plain language I don't want to hear about it. I've a feeling Dick could have done it lying on his couch drinking orange juice. He was truly the last 'common sense' genius. Maybe somewhere up there Fermat and Feyman are laughing their heads off and seeing our foolishness. | |
| DATE | Sunday, May 21st 2006 - 09:40:22 AM | |