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NAME Jess P
E-MAIL jessp54@hushmail.com

COMMENTS

 Excellent web site, Thanks


Jess
DATE Saturday, February 6th 2010 - 12:55:22 PM

NAME BJ Edwards CPFA
E-MAIL 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
E-MAIL 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
E-MAIL 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
E-MAIL 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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DATE Monday, October 19th 2009 - 12:26:45 AM

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DATE Sunday, September 13th 2009 - 01:46:11 PM

NAME Harold
E-MAIL 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
E-MAIL 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
E-MAIL ozirizs@hotmail.com

COMMENTS

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DATE Saturday, August 1st 2009 - 04:01:48 PM

NAME Arnold Madrid
E-MAIL 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
E-MAIL 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
E-MAIL 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
E-MAIL 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
E-MAIL cLargeZoo@yahoo.com

COMMENTS

 Most impresssive!
DATE Saturday, June 27th 2009 - 02:00:09 PM

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DATE Wednesday, June 17th 2009 - 07:26:15 PM

NAME manjunath bs
E-MAIL abc123.m@gmail.com

COMMENTS

 excellent..!
DATE Tuesday, January 6th 2009 - 10:20:24 AM

NAME Parviz Moarefi
E-MAIL pmoarefi@bellsouth.net

COMMENTS

 Simple and to the point. Well done.
DATE Thursday, December 4th 2008 - 08:48:44 AM

NAME Oscar Lindo
E-MAIL 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
E-MAIL inba_jey@yahoo.com

COMMENTS

 i want to read the book
DATE Tuesday, November 11th 2008 - 12:33:14 AM

NAME Jakub Tatomirj
E-MAIL kubana44@yahoo.co.uk

COMMENTS

 Greatings to all of yee
DATE Saturday, October 25th 2008 - 06:03:06 AM

NAME Julita Santos
E-MAIL 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
E-MAIL 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
E-MAIL JIVANNATALARAY@GMAIL.COM

COMMENTS

 no comment
DATE Saturday, August 30th 2008 - 11:04:04 PM

NAME J. IVAN NATALARAY
E-MAIL JIVANNATALARAY@GMAIL.COM
DATE Saturday, August 30th 2008 - 11:02:42 PM

NAME Greg
E-MAIL wyzgreg@rogers.com
DATE Monday, June 23rd 2008 - 07:57:54 AM

NAME FlashDiaz
E-MAIL 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˛
E-MAIL 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
E-MAIL sanam718@yahoo.com

COMMENTS

 a history of time from black holes
DATE Wednesday, May 14th 2008 - 11:41:22 AM

NAME Berrier Dale Byrom
E-MAIL dbyrom51@plateautel.net
DATE Saturday, May 3rd 2008 - 08:58:51 AM

NAME Karsten Tjugen
E-MAIL 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
E-MAIL 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
E-MAIL 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
E-MAIL 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
E-MAIL 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
E-MAIL 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
E-MAIL 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

[Sign] [View] [A Short-Form Proof of Fermat's Last Theorem]

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