21. Notes On Fermat's Last Theorem Alf van der Poorten (Wiley, 1996). Contents, reviews. http://www.maths.mq.edu.au/~alf/NotesonFLT.html

22. Fermat's Last Theorem fermat s last theorem. the proof is not there. To some extent, provingfermat s theorem is like climbing Everest. If a man wants to http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Fermat's_last_theorem.html

Fermat's last theorem Number Theory Index History Topics Index Pierre de Fermat died in 1665. Today we think of Fermat as a number theorist, in fact as perhaps the most famous number theorist who ever lived. It is therefore surprising to find that Fermat was in fact a lawyer and only an amateur mathematician. Also surprising is the fact that he published only one mathematical paper in his life, and that was an anonymous article written as an appendix to a colleague's book. There is a statue of Fermat and his muse in his home town of Toulouse: (Click it to see a larger version) Because Fermat refused to publish his work, his friends feared that it would soon be forgotten unless something was done about it. His son, Samuel undertook the task of collecting Fermat 's letters and other mathematical papers, comments written in books, etc. with the object of publishing his father's mathematical ideas. In this way the famous 'Last theorem' came to be published. It was found by Samuel written as a marginal note in his father's copy of Diophantus 's Arithmetica Fermat's Last Theorem states that x n y n z n has no non-zero integer solutions for x y and z when n Fermat wrote I have discovered a truly remarkable proof which this margin is too small to contain.

23. Fermat's Theorem: DISPROVED fermat s Last theorem Disproved by I. Savant of Marietta, Georgia. Innovativethinking led least an Emmy. fermat s theorem DISPROVED. http://home.mindspring.com/~jbshand/ferm.html

FERMAT's Last Theorem Disproved by I. Savant of Marietta, Georgia. Innovative thinking led to the discovery of solutions to the infamous equation that has baffled mathematicians for a decade. Savant has already become a semi-celebrity, and is the odds on favorite as next years Nobel Prize winner, or at least an Emmy Fermat's Theorem: DISPROVED (IP-Atlanta) The Mathematics community was stunned early yesterday after one of the all-time greatest mysteries was resurrected and then finally put to rest by a Marietta Georgia man. I.Savant, a reclusive bachelor who some say might be related to Elvis or Phyllis Diller, announced that he had discovered several solutions to the what some have called the Holy Grail of Mathematics: Fermat's Last Theorem The theorem is deceiving in its simplicity. Thousands of weeks ago, it was born when the famous mathematician Fermat scribbled a cryptic note in the margin of a journal. The note said that he had stumbled upon a marvelous proof of the following: Unfortunately, as legend has it, Fermat never actually put the proof on paper, and it was lost forever. Every great mathematical mind since has attempted to prove the theorem, and some even claimed success. But Mr. Savant thinks that Fermat knew it would never be proven. Says Mr. Savant: I think Fermat succumbed to pressure when he claimed that he had found a proof, and I don't blame him. I mean, there's this theorem named after you, and they even tell you that it's the last one you're getting. Hell yeah, you're going to tell them you proved it. For years people have tried to show that Fermat's Last Theorem is true. Some have tried to show it was not untrue, and others have tried to show that it was not-not-not unfalse. It dawned upon me that no one had really tried to show that it was un-not not-not-anti-not untrue. When I looked at it this way, I immediately found that it was what I just said it was, and at that point I knew I had stumbled upon a great discovery.

24. Math Forum: Ask Dr. Math FAQ: Fermat's Last Theorem is very indirect, and involves two branches of mathematics which at face value appearto have nothing to do either with each other or with fermat s theorem. http://mathforum.org/dr.math/faq/faq.fermat.html

Ask Dr. Math: FAQ F ermat's L ast T heorem Dr. Math FAQ Classic Problems Formulas Search Dr. Math ... Dr. Math Home What is the current status of Fermat's Last Theorem? In the margin of his copy of a book by Diophantus, Pierre de Fermat wrote that it is possible to have a square be the sum of two squares, but that a cube can not be the sum of two cubes, nor a fourth power be a sum of two fourth powers, and so on. Further, he wrote that he had found a truly marvelous proof which the margin was too small to contain. Fermat's Last Theorem states that x n + y n = z n That is to say, there are no integers x, y, z such that x + y = z , or integers x, y, z such that x + y = z Although this is easily stated, it has proved to be one of the most puzzling problems in the whole history of mathematics. Long after all the other statements made by Fermat had been either proved or disproved, this remained; hence it is called Fermat's Last Theorem (actually, Conjecture would be more accurate than Theorem). This conjecture was worked on by many famous mathematicians. Fermat himself proved this theorem for n = 4, and Leonhard Euler did n = 3. Special cases were dispatched one after another. New theories were developed to attack the problem, but all attempts at a general proof failed. They failed, that is, until this decade, when, building on work of many famous mathematicians, Prof. Andrew Wiles of Princeton University finally proved it. His method could not have been known to Fermat. Fermat's "truly marvelous proof" is now believed to have been faulty.

25. Fermat's Last Theorem Is Solved An attempted elementary proof of FLT using binomial expansions. http://www.coolissues.com/mathematics/fermat.htm

PROOF OF FERMAT'S LAST THEOREM James Constant math@coolissues.com Fermat's Last Theorem is solved using the binomial series Moved to http://www.coolissues.com/mathematics/Fermat/fermat.htm

26. Fermat's Theorem & Negation Of Axiom Of Choice By Ajebara fermat s theorem Negation of Axiom of Choice by ajebara. reply to thismessage post a message on a new topic Back to sci.math.symbolic http://mathforum.org/epigone/sci.math.symbolic/smangjoutheh

reply to this message post a message on a new topic Back to sci.math.symbolic Subject: Author: ajebara@yahoo.com Organization: Deja News - The Leader in Internet Discussion Date: http://www.dejanews.com/rg_mkgrp.xp Create Your Own Free Member Forum The Math Forum

27. Introduction On Bernoulli's Numbers A web article with a brief history and account of their relationship with the Riemann zeta function and fermat's Last theorem (HTML/PS). http://numbers.computation.free.fr/Constants/Miscellaneous/bernoulli.html

Introduction on Bernoulli's numbers (Click here for a Postscript version of this page.) Introduction Bernoulli's numbers play an important and quite mysterious role in mathematics and in various places like analysis, number theory and differential topology. They first appeared in Ars Conjectandi , page 97, a famous (and posthumous) treatise published in 1713, by Jakob Bernoulli (1654-1705) when he studied the sums of powers of consecutive integers s p (n)= n k=1 k p where p and n are two given positive integers. Bernoulli's numbers also appear in the computation of the numbers z k=1 k and in the expansion of many usual functions as tan(x), tanh(x), 1/sin(x), Perhaps one of the most important result is Euler-Maclaurin summation formula, where Bernoulli's numbers are contained and which allows to accelerate the computation of slow converging series (see the essay on Euler's constant at [ ]). They also appear in numbers theory (Fermat's theorem) and in many other domains and have caused the creation of a huge literature (see the 2700 and more entries enumerated in [

28. CATHOLIC ENCYCLOPEDIA: Augustin-Louis Cauchy (Catholic Encyclopedia) Theory of polyhedra, symmetrical functions, proof of a theorem of fermat which had baffled mathematicians like Gauss and Euler. http://www.newadvent.org/cathen/03457a.htm

Home Encyclopedia Summa Fathers ... C > Augustin-Louis Cauchy A B C D ... Z Augustin-Louis Cauchy Napoleon at Cherbourg. While here he devoted his leisure moments to mathematics. Several important memoirs from his pen, among them those relating to the theory of polyhedra, symmetrical functions, and particularly his proof of a theorem of Fermat which had baffled mathematicians like Gauss and Euler, made him known to the scientific world and won him admittance into the Academy of Sciences. At about the same time the Grand Prix offered by the Academy was bestowed on him for his essays on the propagation of waves. After a sojourn of three years at Cherbourg his health began to fail, and he resigned his post to begin at the age of twenty-two his career of professor at the Ecole Polytechnique. In 1818 he married Mlle. de Bure, who, with two daughters, survived him. Napoleon III in the cases of Cauchy and Arago, and he was thus free to continue his lectures. He spent the last years of his life at Sceaux, outside of Paris, devoting himself to his mathematical researches until the end. Society of Jesus VALSON

29. Fermat's Last Theorem From MathWorld fermat's Last theorem from MathWorld A theorem first proposed by fermat in the form of a note scribbled in the margin of his copy of the ancient Greek text Arithmetica by Diophantus. The http://rdre1.inktomi.com/click?u=http://mathworld.wolfram.com/FermatsLastTheorem

30. Fermat's Little Theorem -- From MathWorld fermat s Little theorem. (2). and d divides . Hence,, (3). The theorem is sometimesalso simply known as fermat s theorem (Hardy and Wright 1979, p. 63). http://mathworld.wolfram.com/FermatsLittleTheorem.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Number Theory Congruences Number Theory ... Prime Number Properties Fermat's Little Theorem If p is a prime number and a a natural number , then Furthermore, if p does not divide a ), then there exists some smallest exponent d such that and d divides Hence, The theorem is sometimes also simply known as " Fermat's theorem " (Hardy and Wright 1979, p. 63). This is a generalization of the Chinese hypothesis and a special case of Euler's theorem . It is sometimes called Fermat's primality test and is a necessary but not sufficient test for primality. Although it was presumably proved (but suppressed) by Fermat the first proof was published by Euler in 1749. It is unclear when the term "Fermat's little theorem" was first used to describe the theorem, but it was used in a German textbook by Hensel (1913) and appears is MacLane (1940) and Kaplansky (1945). The theorem is easily proved using mathematical induction on a . Suppose (i.e.

31. Fermat's Last Theorem Video of a popular lecture on FLT arranged by the London Mathematical Society. http://www.lib.ic.ac.uk/av/vids_lms.html#A1.14

32. Carmichael Numbers Test numbers for primality and pseudoprimality in Java. http://www.math.fau.edu/Richman/carm.htm

Primality testing with Fermat's little theorem If n is a prime, and b n , then b n is congruent to 1 modulo n . So if we compute b n modulo n , and don't get 1, then we can conclude that n is not a prime. n A number n is a pseudoprime to the base b if b n is congruent to 1 modulo n . If a number is a pseudoprime to a variety of bases, then it is likely to be a prime. Below you can find out which composite numbers less than m are pseudoprimes to various bases. m A Carmichael number is a composite number n such that b n is congruent to 1 modulo n for every b that is relatively prime to n . So a Carmichael number passes the Fermat's-little-theorem test as best as it can. What are the Carmichael numbers less than m m

33. NOVA Online | The Proof For over 350 years, some of the greatest minds of science struggled to prove whatwas known as fermat s Last theorem the idea that a certain simple equation http://www.pbs.org/wgbh/nova/proof/

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. Text Proof Home Andrew Wiles ... To print NOVA Online is produced for PBS by the WGBH Science Unit

34. Fermat's Little Theorem From MathWorld fermat's Little theorem from MathWorld If p is a prime number and a a natural number, then a^p\equiv a\ \left({{\rm mod\ } {p}}\right). Furthermore, if p\nmid a (p does not divide a), then http://rdre1.inktomi.com/click?u=http://mathworld.wolfram.com/FermatsLittleTheor

35. On A Fermat's Theorem ON A fermat S theorem. Any common factor in two of terms is a factorin all . Let be m the greatest common factor of them. 12 May 1994. http://www.geocities.com/CapeCanaveral/Lab/8266/node2.html

ON A FERMAT'S THEOREM Any common factor in two of terms is a factor in all . Let be m the greatest common factor of them 12 May 1994 Up: CONCEALED PARADOXES IN ...

36. The Prime Glossary: Fermat's Little Theorem Welcome to the Prime Glossary a collection of definitions, information and facts all related to prime numbers. This pages contains the entry titled 'fermat's little theorem.' Come explore a new http://www.utm.edu/research/primes/glossary/FermatsLittleTheorem.html

Fermat's little theorem (another Prime Pages ' Glossary entries) Glossary: Index Search Home Previous ... Mail Editor Prime Pages: Home Search Index Fermat 's "biggest", and also his "last" theorem states that x n + y n = z n has no solutions in positive integers x, y, z with n greater than 2. This has finally been proven by Wiles in 1995. However, in the study of primes it is Fermat's little theorem that is most used: Fermat's Little Theorem. Let p be a prime which does not divide the integer a , then a p mod p It is so easy to calculate a p that most elementary primality tests are built using a version of Fermat's Little Theorem rather than Wilson's Theorem As usual, Fermat did not provide a proof (this time saying "I would send you the demonstration, if I did not fear its being too long" [ Euler first published a proof in 1736, but Leibniz left virtually the same proof in an unpublished manuscript from sometime before 1683. See Also: Fermat quotient Related pages (outside of this work) Proof of this theorem Fermat's theorem, probable-primality and pseudoprimes

37. Instructional Conference On Fermat's Last Theorem University of Illinois at UrbanaChampaign, USA; 618 August 2000. http://www.math.uiuc.edu/~boston/workshop.html

Note: The deadline below passed but there are still a few open spaces. If you or your student is interested, please contact us as soon as possible! Instructional Conference on Fermat's Last Theorem August 6-18, 2000 University of Illinois at Urbana-Champaign Organizing Committee: Nigel Boston UIUC Chris Skinner, IAS and Michigan From August 6-18, 2000, the Instructional Conference on Fermat's Last Theorem will be held as one of the featured events in a Special Year in Number Theory at the University of Illinois. It is intended to provide advanced graduate students with a detailed overview of the recent proof of Fermat's Last Theorem. Workshop participants will arrive Sunday, August 6 and leave at about lunchtime Friday, August 18. The meeting will consist of morning lectures by each of the organizers, followed by breaking into 4 groups of 6 students each to work on projects. These projects will fill some of the holes left in the lectures. Towards the end of the two weeks, students will present talks on their group work. There will be some social events (a reception at the start, an outing in the middle, and banquet at the end). Sponsors The conference is hosted by the Mathematics Department at the University of Illinois and is supported by the Number Theory Foundation and the National Science Foundation.

38. Fermat Biography fermat s theorem s proof finally resolved after several failed attemptfrom so many notable mathematicians who were puzzled for so long. http://math.about.com/library/blfermatbio.htm

zJs=10 zJs=11 zJs=12 zJs=13 zc(5,'jsc',zJs,9999999,'') About Homework Help Mathematics Home ... Math Tutors zau(256,152,145,'gob','http://z.about.com/5/ad/go.htm?gs='+gs,''); Math Help and Tutorials Math Formulas Math Lesson Plans Math Tutors ... Help zau(256,138,125,'el','http://z.about.com/0/ip/417/0.htm','');w(xb+xb); Stay Current Subscribe to the About Mathematics newsletter. Search Mathematics Pierre de Fermat French Number Theorist More Fermat Photos Mathematicians Number Theory Join the Discussion Questions About Fermat? Post Here Forum Membership is Free! Join Today Related Resources Fermat's Last Theorem Proof: Fermat's Theorem Fermat's Proof History of Math Background: P ierre de Fermat (pronounced Fair-mah) was born in Beaumont-de-Lomagne, France in August of 1601 and died in 1665. He is considered to be one of the greatest mathematicians of the seventeenth century. Fermat's father was a leather merchant and his mother's family was in the legal profession. Fermat attended a Franciscan monastery before moving on to obtain a Bachelor's Degree in civil law from the University of Orleans in 1631. He married, had five children and practiced law. For the most part, Math was a hobby for Fermat. Fermat was a busy lawyer and did not let his love of math completely take over his time. It's been said that Fermat never wanted anything to be published as he considered math to be his hobby. The only one thing he did publish - he did so anonymously. He sent many of his papers by mail to some of the best mathematicians in France. It was his link with Marin Mersenne that gave Fermat his international reputation. Fermat loved to dabble in math and rarely provide his proofs (evidence or procedures for reaching conclusions), he would state theorems but neglected the proofs! In fact, his most Famous work 'Fermat's Last Theorem' remained without a proof until 1993 when

39. Richard Taylor's Home Page Publications including the joint paper with Andrew Wiles which completed the proof of fermat's Last theorem. http://www.math.harvard.edu/~rtaylor/

R I C H A R D T A Y L O R Here are some recent papers. They are available either as dvi or as postscript files. They may be very slightly different from the published versions, e.g. they may not include corrections made to the proofs. Galois representations. (Review article.) R.Taylor Proceedings of ICM 2002, volume I, 449-474. dvi Postscript Galois representations. (Long version of above review article.) R.Taylor to appear Annales de la Faculte des Sciences de Toulouse. dvi Postscript Galois representations. R.Taylor slides for talk at ICM 2002. dvi Postscript On the meromorphic continuation of degree two L-functions. R.Taylor preprint. dvi Postscript Remarks on a conjecture of Fontaine and Mazur. R.Taylor Journal of the Institute of Mathematics of Jussieu 1 (2002), 1-19. dvi Postscript On icosahedral Artin representations. II R.Taylor American Journal of Mathematics 125 (2003), 549-566. dvi Postscript On the modularity of elliptic curves over Q. C.Breuil, B.Conrad, F.Diamond and R.Taylor

40. UNC Charlotte Mathematics Department - What We Know About Fermat's Last Theorem A brief history. http://www.math.uncc.edu/flt.php

Chairperson: Dr. Alan Dow Associate Chairperson: Dr. Mohammad Kazemi Coordinator of Graduate Program: Dr. Joel Avrin Coordinator of Undergraduate Program: Dr. Bruno Wichnoski MathEd Coordinator: Dr. Victor Cifarelli Last updated: Back to Main Math Dept. Web Page History of Fermat's Last Theorem Pierre de Fermat (1601-1665) was a lawyer and amateur mathematician. In about 1637, he annotated his copy (now lost) of Bachet's translation of Diophantus' Arithmetika with the following statement: 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. In English, and using modern terminology, the paragraph above reads as: There are no positive integers such that x^n + y^n = z^n for n > 2 . I've found a remarkable proof of this fact, but there is not enough space in the margin [of the book] to write it. Fermat never published a proof of this statement. It became to be known as Fermat's Last Theorem (FLT) not because it was his last piece of work, but because it is the last remaining statement in the post-humous list of Fermat's works that needed to be proven or independently verified. All others have either been shown to be true or disproven long ago.

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