Geometry.Net - Theorems_And_Conjectures: Fermat Theorem

Fermat's Theorem: DISPROVED
fermat s Last theorem Disproved by I. Savant of Marietta, Georgia. For yearspeople have tried to show that fermat s Last theorem is true.
http://home.mindspring.com/~jbshand/ferm.html

Extractions: 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 (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: 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.
Fermat's Last Theorem Is Solved
An attempted elementary proof of FLT using binomial expansions.
http://www.coolissues.com/mathematics/fermat.htm
Fermat's Little Theorem -- From MathWorld
The theorem is sometimes also simply known as fermat s theorem (Hardy and It is unclear when the term fermat s little theorem was first used to
http://mathworld.wolfram.com/FermatsLittleTheorem.html

Extractions: MATHWORLD - IN PRINT Order book from Amazon Number Theory Prime Numbers Primality Testing ... Congruences Fermat's Little Theorem If is a prime number and a natural number , then Furthermore, if does not divide ), then there exists some smallest exponent such that and 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 totient 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 in MacLane (1940) and Kaplansky (1945). The theorem is easily proved using mathematical induction on . Suppose (i.e.
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

Extractions: Introduction on Bernoulli's numbers (Click here for a Postscript version of this page.) 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 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 [
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

Extractions: Home Encyclopedia Summa Fathers ... C > Augustin-Louis Cauchy A B C D ... Z 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. , the well-known preacher, and when, during the reign of Louis-Philippe, the colleges of the Society of Jesus VALSON
Math Forum: Ask Dr. Math FAQ: Fermat's Last Theorem
What is the current status of fermat s Last theorem has it been proved?
http://mathforum.org/dr.math/faq/faq.fermat.html

Extractions: Dr. Math FAQ Classic Problems Formulas Search Dr. Math ... Dr. Math Home 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. 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.
Math Forum - Ask Dr. Math
Why was fermat s theorem such a mystery? Date 01/21/98 at 010830 FromVincent M. Li Subject fermat s theorem I have heard a lot about fermat s
http://mathforum.org/dr.math/problems/li1.21.98.html

Extractions: Associated Topics Dr. Math Home Search Dr. Math Date: 01/21/98 at 01:08:30 From: Vincent M. Li Subject: Fermat's Theorem I have heard a lot about Fermat's Theorem, and how it was once considered one of the world's greatest unsolved mathematical mystries. Supposedly it has been solved by a Professor Andrew Wiles. I was just wondering why it was such a mystey, and how it was proved. I probably won't understand the implications of this for the known world but I am curious to learn. Date: 01/21/98 at 15:20:02 From: Doctor Wilkinson Subject: Re: Fermat's Theorem Here's a webpage with a lot of information: http://www.best.com/~cgd/home/flt/flt01.htm -Doctor Wilkinson, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 01/21/98 at 15:40:21 From: Doctor Rob Subject: Re: Fermat's Theorem You can try the following URL for some of the history of this problem: http://daisy.uwaterloo.ca/~alopez-o/math-faq/node22.html Also go to our Archives search form: http://mathforum.org/mathgrepform.html
Carmichael Numbers
Test numbers for primality and pseudoprimality in Java.
http://www.math.fau.edu/Richman/carm.htm

Extractions: 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.
Flt.html
Hardy and Wright, Davenport, Nagell, , simply use fermat s theorem. And Sierpinski calls it Simple theorem of fermat in his 1964 A Selection of
http://www.spd.dcu.ie/johnbcos/download/Public and other lectures/Fermat's littl

Extractions: The ' little ' of the theorem. When did this theorem start to be called 'Fermat's little theorem? Who (in English) first called it so? Actually not everyone calls it so. In Vol I [1919] of Dickson's monumental three volume [ History of the Theory of Numbers there is an entire chapter devoted to 'Fermat's and Wilson's Theorems.' Hardy and Wright, Davenport, Nagell, ... , simply use 'Fermat's theorem.' And Sierpinski calls it 'Simple Theorem of Fermat' in his 1964 A Selection of problems in the Theory of Numbers Of course everyone knows what 'Wilson's theorem' is - since there is only one such theorem (but, no doubt, someone will write and tell me of another!) - but 'Fermat's theorem'? Well there are several claimants: the beautiful result - to name but one - that every prime p , with (mod 4), is representable by for some (unique; ignoring, of course, change of signs, and interchange) integers a and b , could well claim to be 'Fermat's theorem.' On June 2001 I sent an email to the Number Theory Mailing List enquiring if anyone could answer the above questions. I received several responses (some public, some private), and I will place them in the Fermat's little theorem section of my web site. Briefly, however, I note that a probable answer is that the 'little' came into English from German, but there was no definitive answer as to who first used 'little.'
Fermat Biography
fermat s last theorem and biography. In fact, his most Famous work fermat sLast theorem remained without a proof until 1993 when Andrew J. Wiles
http://math.about.com/library/blfermatbio.htm

Extractions: zJs=10 zJs=11 zJs=12 zJs=13 zc(5,'jsc',zJs,9999999,'') About Homework Help Mathematics Homework Help ... Help zau(256,140,140,'el','http://z.about.com/0/ip/417/C.htm','');w(xb+xb+' ');zau(256,140,140,'von','http://z.about.com/0/ip/496/7.htm','');w(xb+xb); Sign Up Now for the Mathematics newsletter! 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
Fermat's Last Theorem - Wikipedia, The Free Encyclopedia
fermat s last theorem (sometimes abbreviated as FLT and also called fermat s lasttheorem is a generalization of the Diophantine equation a2 + b2 = c2,
http://en.wikipedia.org/wiki/Fermat's_last_theorem

Extractions: Pierre de Fermat Fermat's last theorem (sometimes abbreviated as FLT and also called Fermat's great theorem ) is one of the most famous theorems in the history of mathematics . It states that: The 17th-century mathematician Pierre de Fermat wrote about this in in his copy of Claude-Gaspar Bachet 's translation of the famous Arithmetica of Diophantus : "I have discovered a truly remarkable proof of this theorem that the margin of this page is too small to contain". (Original Latin : "Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caperet.") However, no correct proof was found for 357 years. This statement is significant because all the other theorems proposed by Fermat were settled, either by proofs he supplied, or by rigorous proofs found afterwards. Mathematicians were long baffled, for they were unable either to prove or to disprove it. The theorem was therefore not the last that Fermat conjectured, but the
Fermat S Theorem On Sums Of Two Squares - Wikipedia, The Free
fermat announced this theorem in a letter to Mersenne dated December 25, 1640;for this reason See proofs of fermat s theorem on sums of two squares
http://en.wikipedia.org/wiki/Fermat's_theorem_on_sums_of_two_squares
UNC Charlotte Mathematics Department - What We Know About Fermat's Last Theorem
A brief history.
http://www.math.uncc.edu/flt.php

Extractions: Back to Main Math Dept. Web Page 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.
The Prime Glossary: Fermat's Little Theorem
Welcome to the Prime Glossary a collection of definitions, information and factsall related to prime numbers. This pages contains the entry titled
http://primes.utm.edu/glossary/page.php?sort=FermatsLittleTheorem
Instructional Conference On Fermat's Last Theorem
University of Illinois at UrbanaChampaign, USA; 618 August 2000.
http://www.math.uiuc.edu/~boston/workshop.html

Extractions: 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). 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.
Primality Proving 2.2: Fermat, Probable-primality And Pseudoprimes
fermat s (Little) theorem If p is a prime and if a is any integer, fermat s theorem gives us a powerful test for compositeness Given n 1,
http://primes.utm.edu/prove/prove2_2.html

Extractions: 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 ]. What concerns us here is his "little" theorem: Fermat's (Little) Theorem: If p is a prime and if a is any integer, then a p = a (mod p ). In particular, if p does not divide a , then a p = 1 (mod p proof Fermat's theorem gives us a powerful test for compositeness: Given n a a n modulo n (there is a very easy way to do quickly by repeated squaring, see the glossary page " binary exponentiation "). If the result is not one modulo n , then n is composite. If it is one modulo n , then n might be prime so n is called a weak probable prime base a (or just an a -PRP ). Some early articles call all numbers satisfying this test pseudoprimes, but now the term pseudoprime is properly reserved for composite probable-primes. The smallest examples of pseudoprimes (composite PRPs) are the following. (There are more examples on the glossary page " probable prime ".)
Karl Rubin
Slides for a talk by Karl Rubin on the story of fermat's Last theorem for a general audience, including the history of the problem, the story of Andrew Wiles' solution and the excitement surrounding it, and some of the many ideas used in his proof.
http://math.Stanford.EDU/~rubin/lectures/fermatslides/
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/
NOVA Online | The Proof | Solving Fermat: Andrew Wiles
AW I realized that anything to do with fermat s Last theorem generates too much My wife had heard of fermat s Last theorem, but at that time she had no
http://www.pbs.org/wgbh/nova/proof/wiles.html

Extractions: Andrew Wiles devoted much of his entire career to proving Fermat's Last Theorem, the world's most famous mathematical problem. In 1993, he made front-page headlines when he announced a proof of the problem, but this was not the end of the story; an error in his calculation jeopardized his life's work. Andrew Wiles spoke to NOVA and described how he came to terms with the mistake, and eventually went on to achieve his life's ambition. ANDREW WILES: I grew up in Cambridge in England, and my love of mathematics dates from those early childhood days. I loved doing problems in school. I'd take them home and make up new ones of my own. But the best problem I ever found, I found in my local public library. I was just browsing through the section of math books and I found this one book, which was all about one particular problem Fermat's Last Theorem. This problem had been unsolved by mathematicians for 300 years. It looked so simple, and yet all the great mathematicians in history couldn't solve it. Here was a problem, that I, a ten year old, could understand and I knew from that moment that I would never let it go. I had to solve it.
Fermat Corner
fermat's Last theorem by Simon Singh. Discusses the early and recent history of people trying to solve this perplexing problem, including Andrew Wiles' final success. Includes information about poems, limericks, the offBroadway show and a quiz.
http://www.simonsingh.net/Fermat_Corner.html

Extractions: Fermats Last Theorem dominated my own life for four years, because I made a TV documentary, wrote a book and then lectured on the subject. Getting involved in Fermats mischievous conundrum set me on the path towards being an author and ignited an interest in mathematics that has continued ever since. As a physicist, I was always interested in mathematics as a tool for studying the universe, but learning about Fermats Last Theorem taught me to love mathematics for its own sake. There is a Mathematics Corner currently being developed for this site.

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