Geometry.Net - Theorems_And_Conjectures: Fermat Theorem

Fermat's Last Theorem And The Fourth Dimension
fermat's Last theorem and the Fourth Dimension. This is the webpagefor the article fermat's Last theorem and the Fourth Dimension
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, 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.
Math 491: Fermat's Last Theorem In Context
Math 491, Lecture 1 fermat's Last theorem in Context. (a seminarcourse taught by Prof. James Propp, offered Fall 1999) Schedule
http://www.math.wisc.edu/~propp/courses/491.html

Extractions: (a seminar course taught by Prof. James Propp, offered Fall 1999) Schedule: Tuesdays and Thursdays, 1:00-2:15 pm, Room B223 Van Vleck (note room change!). In the seventeenth century, French mathematician Pierre de Fermat claimed to have proved a result of unprecedented scope, but he did not divulge his proof. Professional and amateur mathematicians alike wrestled with this problem for hundreds of years. A proof was finally found by Princeton mathematician Andrew Wiles in 1994. Why do so many of us mathematicians think Fermat's proof must have been invalid, and why did it take so long for us to find a valid proof? How does the problem fit in among the accomplishments and ambitions of modern mathematics, and what can the story of Fermat's Last Theorem tell us about the nature of intellectual progress in general? In this discussion-based seminar, students will read selections from published books and articles on number theory and the history of mathematics, along with material from a book-in-progress by Prof. Propp directed toward the non-mathematical reading public. We will also watch videos (some popular and some technical). Students will be expected to participate in seminar discussions and to submit weekly one-page written responses to the reading, as well as a final paper or project. Prior knowledge of elementary number theory might be helpful but is not required. Willingness to argue about ideas is essential.
Untitled Document
Dipthantus problem, that fermat's theorem is written next to, does not use integersolutions of x. fermat's theorem, written in Latin, does not use the word
http://www.draaisma.net/rudi/discentries.html

Extractions: X^n + Y^n = Z^n in which n a power greater than 2. However, Diophantus' problem that Fermat wrote his theorem next to does not use integer solutions. In addition, Fermat's Latin text does not use the word whole numbers or integers to describe the solutions. It seem clear that Diophantus and Fermat have both tried to state that the solutions are not integers. Entry sent on Wed 11th Dec 06:17 (GMT) Name: ben Ito Fermat Last Theorem Ben Ito 12-10-02 I will prove that Fermat last theorem is interputed in error. Diophantus' problem, that Fermat wrote the text of his theorem, uses fraction solutions and Fermat's Latin text does not use the word whole numbers or integers to describe the solutions. The whole number solutions of X, Y and Z are not part of Fermat's original Latin text.
Crank Dot Net | Fermat's Last Theorem
fermat's Last theorem Proved and Award Offered for Refutation 2002 May 13 fermat's Last theorem fermat's theorem Disproved 2000 Jul 02
http://www.crank.net/fermat.html

Extractions: Fermat's Last Theorem "During the course of studies on the Goldbach Conjecture, using finite methods, what seems to be an elementary proof of Fermat's 'Last' Theorem has been found. Astonishing here is the lucidity of the arguments and immediacy of their logic. Hopefully, by (numeric) application to the so-called 'hard' problems of Number Theory, some manner of agreement (disputation) will arise." Fermat's Last Theorem "Here we will look at another method of simple proof of Fermat's Last Theorem (FLT) which was published in the booklet 'Fermat's Last Theorem Proved and Award Offered for Refutation' in 1990 with a supplement in 1994 which discussed many invalid but interesting criticisms. (This page will be available on the Internet for interest of mathematicians in seeing the proof, however for the Award conditions ( award valid till the end of 2003) and full discussion of criticisms readers are requested to Purchase the book.)"
Interact On KeelyNet Mail List: Fermat's Theorem A Key To The Unified Field Theo
fermat's theorem a key to the unified field theory? Jerry W.Decker ( (no email) ) Thu, 02 Dec 1999 095643 0600
http://www.keelynet.com/interact/archive/00001687.htm
BletchleyPark.net
fermat's theorem. Congruent Modulo. fermat's Last theorem. Since we're talkingabout fermat, it is suffice to talk about his famous last theorem.
http://www.bletchleypark.net/computation/fermats.html

Extractions: 1 * (6^5) mod(13) = 2 This example can be done on Calc.exe that comes with the MS Windows operating system. First, you can confirm this by doing 6^17 mod(13) and the answer will be 2. However, using Fermat's theorem you can obtain the exponent of 12 from 13 - 1 resulting in 6^12 * 6^5 = 6^17. So, according the Fermat's theorem a^(p - 1) mod p = 1, we have (6^12) mod(13) = 1, which leaves something easily calculable 6^5 mod(13) = 2. Let's now look at a very large exponent: (5^2003) mod(7)
Fermat's Little Theorem
fermat's Little theorem. The theorem is now known as the fermat's Littletheorem to distinguish it from the fermat's Last or Great theorem.
http://www.cut-the-knot.com/blue/Fermat.shtml

Extractions: Recommend this site It comes from observation of multiplication tables Euclid's Proposition VII.30 If two numbers, multiplied by one another make some number, and any prime number measures the product, then it also measures one of the original numbers. p p p , ..., [(p-1)a] p are all different. Which, in terms of remainders, claims that in the sequence no two numbers are congruent modulo p. Assume the opposite: let there be two numbers 1 Proposition VII.30 p p p , ..., [(p-1)a] p permutation p p p , ..., [p-1] p multiplication tables are just permutations of the first row. If two sets are permutations of each other, then products of their elements are clearly equal: [(p-1)!] p p p p p = [a] p p p p = [a p-1 (p-1)!] p = [a p-1 p p Now, dividing by [(p-1)!] p (which is not by Euclid's Proposition VII.30 ) gives 1 = [a p-1 p . Or, in terms of remainders, a p-1 = 1 (mod p) Going over the proof we may notice that it's an overkill to require a to be less than p. The proof remains valid for any a not divisible by p. The statement first appeared without proof in a letter dated October 18, 1640 that Fermat wrote to Frenicle de Bessy . The first proof was given by Leibniz (1646-1716) and the one above was found by Ivory in 1806. Euler proved the theorem in 1736 and its generalization in 1760. The theorem is now known as the Fermat's Little Theorem to distinguish it from the Fermat's Last or Great Theorem. The latter has been finally established by the Princeton mathematician Andrew Wiles (with assistance from Richard Taylor) in 1994.
Marilyn Explains Fermat's Last Theorem
Marilyn Explains fermat's Last theorem. So the question is, was the meaning offermat's theorem changed when Wiles proved it by using Hyperbolic geometry?
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.
Singh, Simon. Fermat's Enigma.
German industrialist vowed to commit suicide at midnight, only to go past his deadlinebecause he had been completely absorbed by working on fermat's theorem.
http://www.ala.org/booklist/v94/adult/oc1/16singh.html

Extractions: How to subscribe to Booklist Magazine Singh, Simon. Fermat's Enigma: The Quest to Solve the World's Greatest Mathematical Problem. Oct. 1997. 288p. index. illus. Walker, $22 (0-8027-1331-9). DDC: 512. For conveying the passions excited by numbers, Singh has chosen a wonderful subject in Fermat's Last Theorem. First, the theorem is easy to grasp, and it has mystery: Fermat in the 1630s wrote of a "marvelous demonstration" of the theorem but didn't reveal what it was. It has tragicomedy: a math-obsessed German industrialist vowed to commit suicide at midnight, only to go past his deadline because he had been completely absorbed by working on Fermat's theorem. Instead of killing himself, he established a prize for solving the theorem. It has romantic tragedy: because of girl troubles, one mathematician died in a duel. Even with all this rich raw material, Singh still had to refine the ore into good writing, and the end product is quite transfixing. Singh begins with Pythagoras himself, to whom numbers were a religion. A faint echo of that ethos is audible throughout Singh's narrative, for once a theorem has been proved, it is true and perfect forever. A Nova broadcast this October will amplify this subject.
Timeline Of Fermat's Last Theorem
Taylor). Timeline of fermat's Last theorem. when, who, what. 1900 Frey'sreasoning Suppose that fermat's Last theorem is not true. Then for
http://www.public.iastate.edu/~kchoi/time.htm

Extractions: Drink to Me (Carolan, sequenced by Barry Taylor) when who what 1900 BC Babylonians A clay tablet, now in the museum of Columbia University, called Plimpton 322, contains 15 triples of numbers. They show that a square can be written as the sum of two smaller squares, e.g., 5 circa 530 Pythagoras Pythagoras was born in Samos. Later he spent 13 years in Babylon, and probably learned the Babylonian's results, now known as the Pythagorean triples. Pythagoras was also the founder of a secret society that studied among others "perfect" numbers. A perfect number is one that is the sum of its multiplicative factors. For instance, 6 is a perfect number (6 = 1 + 2 + 3). Pythagoreans also recognized that 2 is an irrational number. circa 300 BC Euclid of Alexandria Euclid is best known for his treatise Elements circa 400 BC Eudoxus Eudoxus was born in Cnidos, and became a colleague of Plato. He contributed to the theory of proportions, and invented the "method of exhaustion." This is the same method employed in integral calculus. circa 250 AD Diophantus of Alexandria Diophantus wrote Arithmetica , a collection of 130 problems giving numerical solutions, which included the Diophantine equations , equations which allow only integer solutions (e.g, ax + by = c, x
Bigchalk: HomeworkCentral: Fermat's Little Theorem (Discrete Mathematics)
Looking for the best facts and sites on fermat's Little theorem? fermat's lasttheorem; Mathematics behind the theorem; More about fermat's theorem;
http://www.bigchalk.com/cgi-bin/WebObjects/WOPortal.woa/Homework/High_School/Mat
Fermat
fermat Last theorem Ben Ito 1029-02. The whole number solutions is theessence of the modern interputation of fermat's theorem. l. Proof.
http://www.wellstonforum.com/4d/messages/1070.html

Extractions: I will prove that Fermat last theorem is translated in error. Diophantus' problem uses non-whole number solutions and Fermat's Latin text does not use the word whole numbers or integers to describe the solutions. The whole number solutions is the essence of the modern interputation of Fermat's theorem. l. Proof. This is the problem in Arithmetica by Diophantus that Fermat's last theorem was written in the margin. "II.8 To divide a given square number into two squares. Given square number 16. x^2 one of the required squares. Therefore 16 - x^2 must be equal to a square. Take a square of the form (mx - 4)^2, m being any integer and 4 the number which is the square root of 16, e.g., take (2x - 4)^2, and equate it to 16 - x^2. Therefore, 4x^2 - 16x + 16 = 16 - x^2, or 5x^2 = 16x, and x = 16/5. The required squares are therefore 256/25, 144/25." The solutions (x) of this problem are not integers. The only reference to an integer is "m being any integer"; however, (m) is not a solution of the problem. Fermat wrote: Cubem autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere.Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caparet.
Re: Fermat
fermat Last theorem Ben Ito 1029-02. The whole number solutions is theessence of the modern interputation of fermat's theorem. l. Proof.
http://www.wellstonforum.com/4d/messages/1084.html

Extractions: > I will prove that Fermat last theorem is translated in error. Diophantus' problem uses non-whole number solutions and Fermat's Latin text does not use the word whole numbers or integers to describe the solutions. The whole number solutions is the essence of the modern interputation of Fermat's theorem. > l. Proof. > This is the problem in Arithmetica by Diophantus that Fermat's last theorem was written in the margin. "II.8 To divide a given square number into two squares. Given square number 16. x^2 one of the required squares. Therefore 16 - x^2 must be equal to a square. Take a square of the form (mx - 4)^2, m being any integer and 4 the number which is the square root of 16, e.g., take (2x - 4)^2, and equate it to 16 - x^2. Therefore, 4x^2 - 16x + 16 = 16 - x^2, or 5x^2 = 16x, and x = 16/5. The required squares are therefore 256/25, 144/25." > The solutions (x) of this problem are not integers. The only reference to an integer is "m being any integer"; however, (m) is not a solution of the problem.
Euler's Contribution To Number Theory
By Jamie Bailey, April 1999.Category Science Math Number Theory History a prime number. Using this and fermat's theorem, the Eulerfermattheorem can be derived as follows fermat's theorem (fermat's
http://sweb.uky.edu/~jrbail01/euler.htm

Extractions: e, i, f(x), and sigma for summations. He also made significant contributions to differential calculus, mathematical analysis, and number theory, as well as optics, mechanics, electricity, and magnetism. Euler developed the function, which is defined as the number of positive integers not exceeding m that are relatively prime to m. For example, would equal: with(numtheory); phi(7); So, when p is a prime number. Using this and Fermat's Theorem, the Euler-Fermat Theorem can be derived as follows: Fermat's Theorem (Fermat's Little Theorem) states if p is prime and a is a natural number, then ). If p does not divide a , then there exists a smallest d such that ) and d divides p - 1 . Therefore, ). Since when p is prime, the Euler-Fermat Theorem states that , if . In order to prove the Euler-Fermat Theorem, it is necessary to prove the first assertion that ), when p is prime and a is a natural number. So, the problem is to prove Fermat's Little Theorem. Proof: Suppose . It is necessary to show p )). Using the Binomial Theorem
Horizon - Fermat's Last Theorem
fermat's Last theorem. was broadcast on 15 January 1996. At fermat'sLast theorem had baffled mathematicians for over 300 years. But
http://www.maths.nott.ac.uk/personal/ibf/bbcfer.html

Extractions: At the age of ten, browsing through his public library, Andrew Wiles stumbled across the world's greatest mathematical puzzle. Fermat's Last Theorem had baffled mathematicians for over 300 years. But from that day, little Andrew dreamed of solving it. Tonight's HORIZON tells the story of his obsession, and how, thirty years later, he gave up everything to achieve his childhood dream. Deep in our classroom memories lies the enduring notion that "the square of the hypotenuse is equal to the sum of the squares of the other two sides": Pythagoras's Theorem for right-angled triangles. Written down, it is also the simplest of mathematical equations: x + y = z In 1637, a French mathematician, Pierre de Fermat said that this equation could not be true for x + y = z or for any equation x n + y n = z n where n is greater than 2. Tantalisingly, he wrote on his Greek text: "I have discovered a truly marvellous proof, which this margin is too narrow to contain." No one has found the proof, and for 350 years attempts to prove "F.L.T." attracted huge prizes, mistaken and eccentric claims, but met with failure. Simon Singh and John Lynch's film tells the enthralling and emotional story of Andrew Wiles. A quiet English mathematician, he was drawn into maths by Fermat's puzzle, but at Cambridge in the '70s, FLT was considered a joke, so he set it aside. Then, in 1986, an extraordinary idea linked this irritating problem with one of the most profound ideas of modern mathematics: the Taniyama-Shimura Conjecture, named after a young Japanese mathematician who tragically committed suicide. The link meant that if Taniyama was true then so must be FLT. When he heard, Wiles went after his childhood dream again.
DOI 10.1070/im1999v063n05ABEH000262 Citation VA Kolyvagin, On
On the first case of fermat's theorem for cyclotomic fields. Citation, VA Kolyvagin, On the first case of fermat's theorem for cyclotomic fields , Izv.
http://www.turpion.org/php/paper.phtml?journal_id=im&paper_id=262
The British Council Hong Kong - Science - Science Posters - Maths And The World
Maths and the world fermat's theorem. Maths and the world - fermat's theorem InJune 1993 a British Mathematician, Andrew Wiles, proved fermat's theorem.
http://www.britishcouncil.org.hk/science_poster/4.htm

Extractions: Maths and the world - Fermat's theorem Mathematics exists everywhere; in nature, in design, in understanding the physical and chemical processes around us. In June 1993 a British Mathematician, Andrew Wiles, proved Fermat's Theorem. He had studied in the same department at Cambridge University where, in 1998, two mathematicians won the Fields Medal, mathematics' equivalent to the Nobel Prize. Back Produced in Hong Kong by The British Council © 2001. The British Council is the United Kingdom's international organisation for educational and cultural relations. Registered in England as a Charity.
VACETS Technical Column - Tc58
Thu, 1 Dec 1994. fermat's Last theorem (2/2). Last month, we discussedthe fermat's theorem and the prime number stuff and it was fun.
http://www.vacets.org/sfe/fermat2.html

Extractions: "Science for Everyone" "Science for Everyone" was a technical column posted regularly on the VACETS forum. The author of the following articles is Dr. Vo Ta Duc . For more publications produced by other VACETS members, please visit the VACETS Member Publications page or Technical Columns page The VACETS Technical Column is contributed by various members , especially those of the VACETS Technical Affairs Committe. Articles are posted regulary on vacets@peak.org forum. Please send questions, comments and suggestions to vacets-ta@vacets.org Thu, 1 Dec 1994 Fermat's Last Theorem (2/2) Last month, we discussed the Fermat's Theorem and the prime number stuff and it was fun. Slowly, the discussion died out and was replaced with the voting issue. Sad, isn't it? Few days after the [SCIENCE FOR EVERYONE] FERMAT'S LAST THEOREM article was sent out last month, anh Phuong sent the news that the Fermat's last theorem problem was solved by Andrew Wiles and a colleague (I can not remember his name). What I heard at that time was that one of Wiles' colleague said that Wiles' new proof had no hole in it. I haven't heard any more news about that since. I'm wondering if Wiles' new proof has been published, or if the whole mathematical community has accepted the new proof, or if anyone has found new holes in Wiles' new proof. Do any of you hear any thing about that? In that [SCIENCE FOR EVERYONE] FERMAT'S LAST THEOREM article, I had three bonus questions and hoped that someone would be interested in the rewards (pho+? ta`u bay) and would answer them. I did not receive any answer to any of those three questions. It seems that many of us are more interested in politics than science. How sad! Is there a way to change that?
Professor Presents Theorem
Mayo offered a proof of Pierre fermat's theorem using improper and proper evensto TriState academicians and public school representatives during a press
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 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

Extractions: 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

81-100 of 101 Back | 1 | 2 | 3 | 4 | 5 | 6 | Next 20