Geometry.Net - Theorems_And_Conjectures: Fermat Theorem

Math 1013 Website :: Quotes
fermat; I confess that fermat s theorem as an isolated proposition has very little Hilbert On why he didn t try to solve fermat s last theorem
http://ace.acadiau.ca/~jhooper/m1013/quotes.html

Extractions: Gauss Euler Fermat Gauss [A reply to Olbers' attempt in 1816 to entice him to work on Fermat's Theorem.] Gauss Legendre Gauss Hilbert [On why he didn't try to solve Fermat's last theorem] Gauss Euler [upon losing the use of his right eye] Abel [about Gauss' mathematical writing style] Irving Kaplansky Eric Temple Bell Euler Euler Mathematics is a game played according to certain simple rules with meaningless marks on paper. Hilbert Hilbert Barlow Dickson Doob Ramsey Roth Norbert Wiener Hermann Weyl Hermann Weyl Hermann Weyl Von Neumann Titchmarsh Titchmarsh Russell Russell [Upon hearing via Littlewood an exposition on the theory of relativity] Russell Russell a , he says b , he means c , but it should be d Wolfgang Pauli Wolfgang Pauli Mordell Mordell [age 80] Mordell Mordell Mordell Littlewood Littlewood Littlewood Littlewood Littlewood Littlewood Littlewood quotes This page belongs to the math 1013 website ::
Fermat's Last Theorem: Report From A Conference On The Proof By Andrew J. Wiles
fermat s Last theorem Frequently Asked Questions in Mathematics. History offermat s Last theorem What is the current status of FLT?
http://rendezvous.com/tangledweb/conferences/fermat/

Extractions: Back to The Tangled Web FERMAT'S LAST THEOREM Cubem autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere: cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caparet. By contrast it is impossible to separate a cube into two cubes, a fourth power into two fourth powers, or in general any power above the second into two powers of the same degree. I have found a truly marvelous proof of this theorem but this margin is too narrow to contain it. A Conference on the proof of Andrew J. Wiles Boston University August 9-18th, 1995 Interim Reports by Roy Lisker Exclusive to The Tangled Web The Romance of Fermat's Last Theorem Extensive background information about the theorem as well as an analysis of prior "proofs" and excerpts of letters from individuals who claim to have solved Fermat's Last Theorem using only elementary arithmetic. August 11, 1995 The field proves to be a hybrid of number theory and algebraic geometry, with surprisingly little number theory. August 12, 1995
Fermat's Last Theorem And The Fourth Dimension
This is the webpage for the article fermat s Last theorem and the Fourth Dimension , I am also working on a full-length book on fermat s Last theorem,
http://www.math.wisc.edu/~propp/flt4d.html

Extractions: This is the web-page for the article "Fermat's Last Theorem and the Fourth Dimension", presented as a lecture at the fourth Gathering for Gardner in Atlanta, Georgia in February of 2000, and written up shortly afterwards for inclusion in the book "Puzzlers' Tribute: A Feast for the Mind" by David Wolfe (Editor) and Tom Rodgers (Editor), published by A. K. Peters To read the final version of the article (as a Postscript file), click here (If you like my article, buy the whole book! Keep good guys like A. K. Peters in business!) If you have problems with the above file, please let me know! And in the meantime you can fall back on an earlier (rougher) version of the file (created 6/20/00): you have to click here to see the text and here here here here ... here , and here to get the seven individual figures that accompany the text. I plan to turn this page into a hypertex document sometime soon, with links that provide further background information, references, technical caveats, etc.; watch this space for developments. I am also working on a full-length book on Fermat's Last Theorem, entitled "Who Proved Fermat's Theorem?: The Curious Incident of the Boasting Frenchman", to be published by Princeton University Press; I'll post more details here as they become available.
Marilyn Explains Fermat's Last Theorem
If we reject Wiles proof of fermat s Last theorem, must we also reject So the question is, was the meaning of fermat s theorem changed when Wiles
http://www.wiskit.com/marilyn/fermat.html

Extractions: Marilyn is Wrong Herb Weiner by Marilyn vos Savant is a column in Parade Magazine , published by PARADE, 711 Third Avenue, New York, NY 10017, USA. According to Parade, Marilyn vos Savant is listed in the "Guinness Book of World Records Hall of Fame" for "Highest IQ." In her Parade Magazine column of November 21, 1993, Marilyn reports that she does not believe that Andrew Wiles has succeeded in proving Fermat's Last Theorem, because the proof relies on non-Euclidean (hyperbolic) geometry. Note that in the discussion that follows, the notation x**n is used to represent x to the nth power. Just because a tool is inappropriate for one task does not mean that tool is inappropriate for all other tasks. If we reject Wiles' proof of Fermat's Last Theorem, must we also reject Einstein's General Theory of Relativity? When we are asked to solve a problem using "only a ruler and a straightedge," I think it's inappropriate to rely on non-Euclidean geometry. While Wiles' proof is clearly not the same as Fermat's "remarkable" proof, I don't understand why it's invalid simply because it relies on non-Euclidean geometry. As a non-mathemetician, this is the strongest argument I can make. I'll surely be hearing from my creative readers.
PRACTICE WITH POWERS (pre 2nd Midterm) SOLUTIONS 1. Evaluate 3
That s as far as fermat s theorem will take us, so now we quickly check for shortpatterns or early 1 s (or 1 s) in the first few powers of 7 Quick Check
http://www.math.harvard.edu/~sarah/magic/practice/powersols

Extractions: PRACTICE WITH POWERS (pre 2nd midterm) SOLUTIONS 1. Evaluate 3^2343 (mod 19). Answer: 8. Reduction by Fermat's Theorem - Since 19 is prime, Fermat's Theorem says that 3^18 == 1 (mod 19). Thus we take 2343 = 18x130 + 3. 3^2343 == (3^18)^130 x 3^3 == 1 x 27 == 27 == 8 (mod 19). - 2. Evaluate 2^375961 (mod 59). Answer: 32. Reduction by Fermat's Theorem - Since 59 is prime, Fermat's Theorem says that 2^58 == 1 (mod 59). Thus we take 375961 = 58x6482 + 5. 2^375961 == (2^58)^6482 x 2^5 == 1 x 32 == 32 (mod 59). 3. Evaluate 7^352426 (mod 61). Answer: 45. Reduction by Fermat's Theorem - Since 61 is prime, Fermat's Theorem says that 7^60 == 1 (mod 61). Thus we take 352426 = 60x5873 + 46. 7^352426 == (7^60)^5873 x 7^46 == 1 x 7^46 == 7^46 (mod 61). That's as far as Fermat's Theorem will take us, so now we quickly check for short patterns or early 1's (or -1's) in the first few powers of 7: Quick Check for Short Patterns 7^1 == 7, 7^2 == 49, 7^3 == 38, 7^4 == 22 (mod 61). Nope! No obvious short patterns. Next strategy! Successive Squaring for 7^46 (mod 61) - 1) Express 46 as a sum of powers of 2: 46 = 32 + 8 + 4 + 2. 2) Calculate 7^(powers of 2) by successive squaring: 7^1 == 7 == 7 (mod 61) 7^2 == 7^2 == 49 == 49 (mod 61) 7^4 == 49^2 == 2401 == 22 (mod 61) 7^8 == 22^2 == 484 == 57 (mod 61) 7^16 == 57^2 == (-4)^2 == 16 (mod 61) 7^32 == 16^2 == 256 == 12 (mod 61) 3) Combine some of these by multiplication to get 7^46: 7^46 == 7^(32 + 8 + 4 + 2) == 7^32 x 7^8 x 7^4 x 7^2 == 12 x 57 x 22 x 49 == 737352 == 45 (mod 61). Finished! 4. Evaluate 5^33 (mod 28). Answer: 13. 28 is NOT PRIME, so NO FERMAT! Quick Check for Short Patterns 5^1 == 5, 5^2 == 25, 5^3 == 13, 5^4 == 9 (mod 28). Nope! No obvious short patterns. Next strategy! Successive Squaring for 5^33 (mod 28) - 1) Express 33 as a sum of powers of 2: 33 = 32 + 1. 2) Calculate 5^(powers of 2) by successive squaring: 5^1 == 5 == 5 (mod 28) 5^2 == 5^2 == 25 == 25 (mod 28) 5^4 == 25^2 == 625 == 9 (mod 28) 5^8 == 9^2 == 81 == 25 (mod 28) 5^16 == 25^2 == 625 == 9 (mod 28) 5^32 == 9^2 == 81 == 25 (mod 28) 3) Combine some of these by multiplication to get 5^33: 5^46 == 5^(32 + 1) == 5^32 x 5^1 == 25 x 5 == 125 == 13 (mod 28). Finished! 5. Evaluate 9^149 (mod 21). Answer: 18. 21 is NOT PRIME, so NO FERMAT! Quick Check for Short Patterns 9^1 == 9, 9^2 == 18, 9^3 == 15, 9^4 == 9 (mod 21). Yippee! A pattern! We have a cycle of length 3 that goes 9, 18, 15, 9, 18, 15, 9, 18, 15. 149 = 49x3 + 2, so we complete 49 full cycles, and then stop at the 2nd number in the cycle. Thus 9^149 == 9^2 == 18 (mod 21).
QR28 Magic Of Numbers Dick Gross 11/25/02 Finding Roots (mod N) I
Euler s proof of fermat s theorem The following ideas are due to Leonhard Euler,born 1636. Euler is my hero! He was one of the most original
http://www.math.harvard.edu/~sarah/magic/topics/dick
Fermat's Last Theorem Poetry Challenge
The proof of fermat s last theorem by Andrew Wiles and Richard Taylor was presentedto fermat s theorem has been solved, What will now make math evolve?
http://www.math.uic.edu/~jeremy/poetry.html

Extractions: While you're here, you can check out my home page, or visit the UIC math department. The proof of Fermat's last theorem by Andrew Wiles and Richard Taylor was presented to an audience of over 300 people during a tenday conference at Boston University in August, 1995. At that conference, I issued a poetry challenge asking for occasional verse to celebrate the proof. While the authors' anonymity was preserved at the meeting, all things are revealed in time. If you would like to contribute to this poetry competition, please send your masterpiece to Jeremy Teitelbaum . The editor's decisions regarding suitability for publication in this forum are arbitrary, personal, and final. With thanks to all of the participants, here are the entries (in no particular order). Author: John Fitzgerald Fermat's last theorem
38.
fermat has stated that the equation xn + yn = zn cannot be solved by integersexcept for n = 1 and 2; this is called today fermat s theorem.
http://kr.cs.ait.ac.th/~radok/math/mat1/mat138.htm

Extractions: Irrational numbers During the Determination of the square root we have encountered calculations, which we could continue for ever such as the calculation of the You might think that this action will finish or lead to a periodic decimal, so that it could be finished. You would then have found a finite decimal fraction or a periodic infinite decimal fraction , which you could convert into a simple fraction . Hence you might ask whether is equal to a simple fraction. Let p q , where p and q are integers without common factors and q p q p q p p q q Since p q have no common factors and q assumption is wrong The calculation of continues and has no end. Therefore it is a new type of number. So far, we have encountered only integers, common fractions and decimal fractions with finite numbers of decimals or periodic decimals . All these numbers, which can be positive or negative, share the property of being representable as common fractions and are called rational numbers . You might ask whether we should accept the existence of such new numbers. Since mathematics aims to remove as many restrictions as possible, it has accepted these numbers and called them irrational numbers . These numbers cannot be expressed in terms of rational numbers - as common fractions or decimal fractions with finite or periodic decimal digits. You can easily construct such numbers, for example:
Faces Of The Moon
Andrew Wiles claimed in 1993 that he was able to prove fermat s theorem.This announcement made history in the field of mathematics. Or is it?
http://perryv.i.ph/blogs/facesmoon/?&offset=25
Faces Of The Moon
Pinoy s feat on fermat s theorem turns out to be fake Andrew Wiles claimedin 1993 that he was able to prove fermat s theorem.
http://perryv.i.ph/blogs/facesmoon/index.php?category=science-and-trivia&blog=fa
A Flaw In The Proof Of Fermat's Theorem
A flaw in the proof of fermat s theorem whoever knows anything about A FLAWIN THE RECENT PROOF OF fermat S LAST theorem Please inform us!
http://www.seas.upenn.edu/~sweirich/types/archive/1994/msg00002.html

Extractions: [Prev] [Next] [Index] [Thread] [This message is not strictly relevant to the Types Forum. But I am circulating it anyway, as a matter of general interest. Replies by e-mail to Mamede please. Mamede, could you please post a summary or pointer to your final article, for those of us that are interested? Philip Wadler, moderator, Types Forum] Please, whoever knows anything about A FLAW IN THE RECENT PROOF OF FERMAT'S LAST THEOREM: Please inform us! We need to write a report on the situation to a Brazilian magazine and we are looking for fresh news. Thanks a lot, Mamede Lima-Marques Center for Logic and Epistemology UNICAMP Campinas - Brazil Prev: ABSTRACT Next: BRA project `Types for Proofs and Programs' Index(es): Main Thread
Professor Presents Theorem
Mayo offered a proof of Pierre fermat s theorem using improper and proper I have a solution to fermat s theorem that a schoolchild could understand.
http://www.marshall.edu/parthenon/archives/20000912/news/professor.html

Extractions: reporter After more than 300 years, a possible new discovery in the area of mathematics has been made by Marshall student and Ohio University instructor Walter Mayo. Mayo offered a proof of Pierre Fermat's theorem using improper and proper evens to Tri-State academicians and public school representatives during a press conference. "I have solved this formula using a technique that Fermat could have used in the 1600s," Mayo said. "I was trying to put a mathematical structure on prime numbers and noticed a difference in even numbers. "This solution is beautiful because it is so simple," he added. "I have a solution to Fermat's Theorem that a school-child could understand." However, no such explanation was given. This possible discovery of even numbers allowed Mayo to prove Fermat's famous Last Theorem. Mayo said his proof is being examined by a number of technical and non-technical members of the scientific community. In the 1600s, Fermat was working on an extension of the Pythagorean Theorem. Fermat said he had discovered a proof for his extension, but died before he could reveal it and no proof was ever found. "I went to Howard University in Washington, D. C., and showed them my proof," Mayo said. "They were astonished by the simplicity of it. This discovery could lead to a whole new area in number theory.
[FERMAT, Pierre De] DIOPHANTUS Of Alexandria, Arithmeticorum Libri Sex, Et De Li
Of course since the above was written fermat s theorem has been proved by AndrewWiles, with the collaboration of several other mathematicians.
http://www.polybiblio.com/watbooks/2444.html

Extractions: 2 for which the equation xn + yn = zn holds true for the positive integers x, y, and z. Fermat claimed in his note to have discovered a '"very wonderful demonstration" (demonstratio sane mirabilis) of this theorem that the margin was too narrow to contain, a claim that has tantalized mathematicians ever since. The remainder of Fermat's notes contained a large number of theorems on the theory of numbers, only one of which he himself proved; the rest were proven in the eighteenth century' (Norman catalogue). The collation of this copy agrees with Norman and several other copies recorded. However, a very few copies contain a portrait, and occasionally an errata leaf, both of which appear to have been issued later as witnessed by the number of copies in their original bindings, as this one, with no evidence of removal.
Table Of Contents
fermat s theorem. 3. Exponents of numbers. 4. Primitive roots for primes The Converse of fermat s theorem. 1. The converse of fermat s theorem
http://web.doverpublications.com/cgi-bin/toc.pl/0486656209

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Canadians Are Smug - Prodding Fermat With The New Kind Of Science Explorer
fermat s theorem says that if you don t use squares, instead you use, Continuing on with fermat s theorem means looking at some successive powers.
http://smugcanadian.weblogger.com/stories/storyReader$247

Extractions: Sign In Posted by Smug Canadian , 4/24/03 at 12:36:50 AM. I just got the A New Kind of Science Explorer , which is the software accompaniment to Stephen Wolfram's book of the same name. I've got a hunch that Wolfram is onto something, and I would love to see his technique vindicated. Here's my overly ambitious attempt to look at Fermat's Last Theorem. Here's a quick math lesson for those who have forgotten. You probably remember Pythgoras' theorem: Simple enough. Fermat's theorem says that if you don't use squares, instead you use, for example, cubes: After seeing how Wolfram revealed the random complexity of numbers by showing them in binary, I figured there must be a use for this stuff to visualize difficult problems like Fermat's theorem so that it is possible at a glance to see why the theorem is true. I can't really even follow basic mathematical proofs, let alone Wiles' hard math , so I can never write an actual proof. But I am nutty enough to think that there is a way to see the proof that I don't have to understand that stuff. Here goes.
NOVA Online | The Proof
NOVA Online presents The Proof, including an interview with Andrew Wiles, anessay on Sophie Germain, and the Pythagorean theorem.
http://www.pbs.org/wgbh/nova/proof/

Extractions: For over 350 years, some of the greatest minds of science struggled to prove what was known as Fermat's Last Theorem the idea that a certain simple equation had no solutions. Now hear from the man who spent seven years of his life cracking the problem, read the intriguing story of an 18th century woman mathematician who hid her identity in order to work on Fermat's Last Theorem, and demonstrate that a related equation, the Pythagorean Theorem, is true.

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