61. Flt.html simply use 'fermat's theorem.' And Sierpinski calls it 'Simple theorem offermat' in his 1964 A Selection of problems in the Theory of Numbers . http://www.spd.dcu.ie/johnbcos/download/Public and other lectures/Fermat's littl

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.'

62. The Proof Of Fermat's Last Theorem The Proof of fermat's Last theorem. For a quick Proof of fermat's Lasttheorem from the TaniyamaShimura conjecture. After Frey drew http://www.mbay.net/~cgd/flt/flt08.htm

The Proof of Fermat's Last Theorem For a quick definition of many of the terms used here, you may refer to the Glossary External references for this section: [Gou], [Rib], [Rih] Contents: Frey curves Proof of Fermat's Last Theorem from the Taniyama-Shimura conjecture Proof of the semistable case of the Taniyama-Shimura conjecture Frey curves Suppose there were a nontrivial solution of the Fermat equation for some number n, i. e. nonzero integers a, b, c, n such that Then we recall that around 1982 Frey called attention to the elliptic curve Call this curve E. Frey noted it had some very unusual properties, and guessed it might be so unusual it could not actually exist. To begin with, various routine calculations enable us to make some useful simplifying assumptions, without loss of generality. For instance, n may be supposed to be prime and 5. b can be assumed to be even, a 3 (mod 4), and c 1 mod 4. a, b, and c can be assumed relatively prime. The "minimal discriminant" of E, can be computed to be - a power of 2 times a perfect prime power. One unusual thing about E is how large the discriminant is.

63. The Mathematics Of Fermat's Last Theorem That, then, is a very brief overview of the mathematical cast of characterswhich play leading roles in the eventual resolution of fermat's theorem. http://www.mbay.net/~cgd/flt/fltmain.htm

The Mathematics of Fermat's Last Theorem Welcome to one of the most fascinating areas of mathematics. There's a fair amount of work involved in understanding even approximately how the recent proof of this theorem was done, but if you like mathematics, you should find it very rewarding. Please let me know by email how you like these pages. I'll fix any errors, of course, and try to improve anything that is too unclear. Introduction If you have ever read about number theory you probably know that (the so-called) Fermat's Last Theorem has been one of the great unsolved problems of the field for three hundred and fifty years. You may also know that a solution of the problem was claimed very recently - in 1993. And, after a few tense months of trying to overcome a difficulty that was noticed in the original proof, experts in the field now believe that the problem really is solved. In this report, we're going to present an overview of some of the mathematics that has either been developed over the years to try to solve the problem (directly or indirectly) or else which has been found to be relevant. The emphasis here will be on the "big picture" rather than technical details. (Of course, until you begin to see the big picture, many things may look like just technical details.) We will see that this encompasses an astonishingly large part of the whole of "pure" mathematics. In some sense, this demonstrates just how "unified" as a science mathematics really is. And this fact, rather than any intrinsic utility of a solution to the problem itself, is why so many mathematicians have worked on it over the years and have treated it as such an important problem.

64. Sex, Botany And Fermat's Theorem Publication Date Friday Sep 22, 2000. Sex, botany and fermat's theorem. TomStoppard's heady Arcadia opens Bus Barn Stage Company's sixth season http://www.paweekly.com/PAW/morgue/listings/2000_Sep_22.ARCADIA.html

Publication Date: Friday Sep 22, 2000 Sex, botany and Fermat's theorem Tom Stoppard's heady "Arcadia" opens Bus Barn Stage Company's sixth season by Laura Reiley It's about sex. No, wait. It's really about the second law of thermodynamics (you know, the one that says the universe is gradually becoming more, not less, diffuse and chaotic). Oh, and it's about Lord Byron, and botany. But definitely, definitely about sex. Tom Stoppard's "Arcadia" is, like so many of his plays, a potpourri of intellectual ideas, puns, jokes, double entendres, and lots of other eyebrow-raising language shenanigans. But as Barbara Cannon, director of Bus Barn Stage Company's upcoming production of the show notes, "Arcadia" is also a whodunnit. "Stoppard never gives you the answer, but he gives you a lot of clues," Cannon said. "The play about romanticism versus rationalism, the history of landscape architecture, and so forth. But it has a stronger plot (than other Stoppard plays). It will be easier for an audience to follow." "Arcadia" opens tonight at the Bus Barn Stage Company in Los Altos.

65. Fermat's Theorem fermat's theorem. x 2 + y 2 = z 2. Where as fermat's theorem states thatx n + y n = z n. Has no noninteger solutions for x, y and z when n 2. http://www.bath.ac.uk/~ma0drje/www.Fermat/Fermats Theorem.html

Fermat's Theorem Home Page Introduction Fermat Theorem ... Links The problem looks very straight forward, as it is based on a piece of mathematics that everyone can remember; ‘In a right angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides.’ Pythagoras' Theorem in terms of x, y and z is: x + y = z Where as Fermat's theorem states that: x n + y n = z n Has no non-integer solutions for x, y and z when n > 2. In about 1637, he annotated his copy (now lost) of Bachet's translation of Diophantus' Arithmetika with the following statement: 'Cubum 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 caperet.' In English, and using modern terminology, the paragraph above reads as: There are no positive integers such that 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. Although he supposedly proved the theorem most mathematicians now believe that Fermat's 'proof' was wrong although it is impossible to be completely certain. The problem set by Fermat was to;

66. G.Imbalzano On II Fermat'theorem & Structure Fine'constant. Translate this page ABSTRACT.txt ~ Riflessioni sull'Ultimo .. fermat ~. Ricerca DIDATTICA su //services.csi.it/~major/ Lyricae ~ Merci, mon ami! ~. =! Links PREFERITI != ! http://web.infinito.it/utenti/i/imbalzano/newsiol.htm

Associazione Insegnanti di Fisica: XXXV Congresso ~~~Use MS LineDraw (fixed) FONT~~~ ABSTRACT.txt ~ Riflessioni sull'Ultimo .. Fermat ~ Ricerca DIDATTICA su ://services.csi.it/~major/ #Lyricae# ~ Merci, mon ami! ~ =! Links PREFERITI != ! Antigravity Bibliography .. Very LONG ! Home Page on http://www.geocities.com/jmbalzan

67. The Smallest Power Congruent To 1: Fermat's Theorem The smallest power congruent to 1 fermat's theorem. For some further examples, youcan appreciate fermat's theorem by checking , , etc. It really is amazing. http://www.math.okstate.edu/~wrightd/crypt/lecnotes/node19.html

Next: Square roots Up: Powers in Modular Arithmetic Previous: Relatively prime numbers The smallest power congruent to 1: Fermat's theorem We will begin with a prime modulus p . Choose any a from 1 to p -1and consider the multiples Remembering our experience with multiplicative ciphers, we know that these numbers are all different modulo p . That is, modulo p , these are precisely all the numbers from 1 to p Here is the trick! Multiply all the numbers together and the result must be the same as multiplying all the numbers from 1 to p -1. Thus, The left side also has ( p -1)! in it as well. After some rearrangement The prime p does not divide ( p -1)!. That means that we can cancel it from both sides of the congruence, and at last obtain Fermat's Theorem: For any number a relatively prime to the prime p , we have By this theorem, we can conclude immediately that . As a word of caution, it may turn out that a smaller power will produce 1. For example, if a =9, then . So a smaller power of 9 is congruent to 1 mod 19. For some further examples, you can appreciate Fermat's theorem by checking

68. EULER'S FUNCTION; EULER'S THEOREM; FERMAT'S THEOREM http://www.ece.utexas.edu/~prob/OTHER-COURSES/CRYPTO/math.background/node27.html

Next: MULTIPLICATIVE FUNCTIONS Up: Incongruences Previous: SETS OF RESIDUES E ULER S FUNCTION ; E ULER S THEOREM ; F ERMAT S THEOREM In general, we let be the number of elements of . The function is the so-called Euler's (totient) function , where, by definition, . The first thing that we prove about is Euler's theorem m n )=1 then . The special case where n p , a prime, is called Fermat's theorem If p is prime then, for all m To prove Euler's theorem, fix and consider all its powers (under multiplication ). Since is a finite set, there are only finitely many distinct powers. Denote this set by . Let o m ) be the number of elements of ( m ). Obviously, . Now define an equivalence relation: iff for some integer k . Let [ a ] be the equivalence class of a under this relation. It is easy to see that [ a ] has exactly the same number of elements as ( m ), i.e. o m ). Hence is a multiple of o m ). Since , it follows that as well. n . Alternatively, by picking a representative from each class, in a way that this representative is between 1 and n , we think of as a so-called reduced set of residues ) m n )=1, there is a natural one-to-one correspondence between and . This is defined by the function

69. The Laws Of Cryptography: Fermat's Theorem Illustrated The Laws of Cryptography fermat's theorem Illustrated by Neal R. Wagner. Copyright© 2002 by Neal R. Wagner. All rights reserved. fermat's theorem. http://www.cs.utsa.edu/~wagner/laws/AFermat.html

The Laws of Cryptography: Fermat's Theorem Illustrated by Neal R. Wagner Fermat's Theorem. Recall that Fermat's theorem says that given a prime p and a non-zero number a a p-1 mod p is always equal to . Here is a table for p = 11 illustrating this theorem. Notice below that the value is always by the time the power gets to , but sometimes the value gets to earlier. The initial run up to the value is shown in red boldface in the table. A value of a for which the whole row is red is called a generator . In this case , and are generators. Values p a for p a a a a a a a a a a a Java code to produce the table above and the one below. Here is a larger table with p = 23 . There are generators. Values p a for p a a a a a a a a a a a a a a a a a a a a a a a Revision date: . (Please use ISO 8601 , the International Standard.)

70. Ask Jeeves: Search Results For "Fermat's Theorem Wiles Proof" Popular Web Sites for fermat's theorem Wiles Proof . powered by SMARTpages.com.Ask Jeeves a question about fermat's theorem Wiles Proof Search the Web for http://webster.directhit.com/webster/search.aspx?qry=Fermat's Theorem Wiles Proo

71. Review: Fermat's Enigma Despite its simple form, fermat's theorem had defied the attemptsof mathematicians to prove or disprove it for three centuries. http://www.sasquatch.com/~kory/fermats_enigma.html

Fermet's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem Simon Singh Walker and Company NY 1997 ISBN 0-8027-1331-9 Fermat's Enigma is a popular account of the history of Fermat's famous Last Theorem and its proof in 1995 by Andrew Wiles. Singh provides a readable account, including a good deal of background information about various branches of mathematics that intersect the problem. Some of this background is by way of illustrating the nature of mathematical proof, and it is refeshing in this postmodern age to come across a reminder of the insistence on absoluteness in mathematical proof, as that is one the charms of mathematics: "Scientific proof [i.e. as used in physical sciences] is inevitably fickle and shoddy. On the other hand, mathematical proof is absolute and devoid of doubt." (p. 23) Despite its simple form, Fermat's Theorem had defied the attempts of mathematicians to prove or disprove it for three centuries. As is well known, Wiles's actual proof was of a theorem known as the Taniyama-Shimura conjecture, which asserts that certain kinds of equations (known for historical reasons as elliptic curves, or elliptic equations) of the form y = x +ax +bx+c a b , and c being integers, or whole numbers) have a deep identity with certain mathematical objects known as ``modular forms''. Now, I'm not a mathematician, so I know nothing about modular forms aside from what I've read here. Singh's description is somehat brief; to my untrained mind they seem a bit like some sort of vector thingy living in an odd kind of four-dimensional space. The Taniyama-Shimura conjecture says that every elliptic equation (specified by its

72. The Prime Glossary: Partial Index: F fermat's little theorem; fermat's method of factoring; fermat's theorem;fermatCatalan conjecture (updated) fermat-Catalan equation; http://primes.utm.edu/glossary/index.php?match=f

73. The Prime Glossary: Circular Prime all related to prime numbers. This pages contains the entry titled'fermat's little theorem.' Come explore a new prime term today! http://primes.utm.edu/glossary/page.php?random=tetradic

74. Fermat's Theorem NUMBER THEORY. fermat's theorem. If P is a prime and the gcd (a, p) = 1, thena P1 = 1 mod p. Ex. Verify that 3 16 = 1 mod 17. We know 3 3 = 10 (mod 17). http://www.cs.umbc.edu/~stephens/crypto/TOPICS/math.html

NUMBER THEORY Fermat's Theorem If P is a prime and the gcd (a, p) , then a P-1 1 mod p Ex. Verify that 3 = 1 mod 17 We know: 3 = 10 (mod 17) = 100 = 15 (mod 17) = 225 = 4 (mod 17) = 4 * 10 * 3 = 120 =1 mod 17 Ex. Verify that 18 = 1 mod 49 We know: 18 = 324 = 30 (mod 49) = 900 = 18 (mod 49) = 18 * 30 = 1 mod 49 Euler's F - Function F F (n) = n - 1 iff n is prime Ex. F F If n pq and p, q are prime, then F (n) F (p) F (q) p - 1 q-1 Ex. F F F if p is prime and K F (p k p k p k-1 = p k-1 p Ex. F F -or- F F if n p k p k p r kr F (n) p k p k p k p k p r kr p r kr Ex. F F F F Euler's Theorem n a, n ) =1, then a F (n) = 1 (mod n Ex. n F n if a = -3 then, F a * r i a * r I -3 * -6 * -9 * -12 * -15 * -18 = 4 * 1 * 5 * 2 * 6 * 3 (mod 7) 1 (-3) * 2 (-3) * 3 (-3) * 4 (-3) * 5 (-3) * (6) (-3) = 4 * 1 * 5 * 2 * 6 * 3 (mod 7) = 1 * 2 * 3 * 4 * 5 * 6 (mod 7) = 1 (mod 7) Chinese Remainder Theorem Let n , n , ...... n k , be integers such that the gcd (n , n j ) = 1 for i not equal to j. Then the system of linear congruencies: x a r mod( n r ) where r k . has a unique solution modulo n n n k Ex. Find the unique solution of X satisfying the system of simultaneous congruencies: x = 3(mod 5)

75. Primality Proving 2.2: Fermat, Probable-primality And Pseudoprimes fermat's theorem gives us a powerful test for compositeness Given n 1, choosea 1 and calculate a n1 modulo n (there is a very easy way to do quickly by http://www.utm.edu/research/primes/prove/prove2_2.html

2.2: Fermat, probable-primality and pseudoprimes Home Primality Proving Chapter Two 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 ".)

76. Fermat's Little Theorem On this page we give the proof of fermat's Little theorem (a variant of Lagrange'stheorem). fermat's Little theorem (from the Prime site's list of proofs). http://www.utm.edu/research/primes/notes/proofs/FermatsLittleTheorem.html

Fermat's Little Theorem (from the Prime site 's list of proofs Home Search Site Largest ... Submit primes 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 > 2. This has finally been proven by Wiles in 1995. Here we are concerned with his "little" but perhaps his most used theorem which he stated in a letter to Fre'nicle on 18 October 1640: Fermat's Little Theorem. Let p be a prime which does not divide the integer a , then a p = 1 (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" [ , p79]). Euler first published a proof in 1736, but Leibniz left virtually the same proof in an unpublished manuscript from sometime before 1683. Proof. Start by listing the first p -1 positive multiples of a a a a p a Suppose that ra and sa are the same modulo p , then we have r s (mod p ), so the

77. TV STUDIO Videos For Sale - L.M.S Video of a popular lecture on FLT arranged by the London Mathematical Society.Category Science Math Diophantine Equations fermat s Last theorem...... Dr Richard Pinch This video gives a brief history of the attempts that have beenmade to solve fermat's theorem (said to have been his last), and outlines some http://www.lib.ic.ac.uk/av/vids_lms.html

London Mathematical Society - 'Popular Lecture Series' Games Animals Play. Professor J.Maynard Smith Game theory is applied by an eminent biologist to give an insight into animal contests (for instance, for mates), leading to an explanation of why there are an (almost) equal number of male and female births, the behaviour of the Hamadryas Baboon, and the funnel web spider. Prerequisites: none, but a knowledge of pay-off matrix would be helpful and some idea of evolution. prod# 1106 - dur 38'06" - year of production July 1986. Order details The Rise and Fall of Matrices. Professor W.Ledermann. A description of the revolutionary paper of 1858 by Cayley and the change of emphasis of the teaching of linear algebra from determinants to matrices and linear maps, with historical background. Prerequisites: Some knowledge of Matrices. prod# 1107 - dur 42'48" - year of production July 1986. Order details Games That Solve Problems Professor W.A.Hodges. Mathematicians don't just solve problems. They also find methods for solving new kinds of problems. How can they do this? The lecture describes various attempts to answer this question during the last 150 years. One important recent approach is based on a kind of 'spot the difference' game; simple examples are given. Prerequisites: an interest in abstract ideas and patterns.

78. 38. to arbitrary n. fermat has stated that the equation x n + y n = z n cannot be solvedby integers except for n = 1 and 2; this is called today fermat's theorem. http://kr.cs.ait.ac.th/~radok/math/mat1/mat138.htm

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:

79. Abel.math.harvard.edu/~sarah/magic/topics/powersols Answer 8. Reduction by fermat's theorem Since 19 is prime, fermat's theorem says that 3^18 1 (mod 19). http://abel.math.harvard.edu/~sarah/magic/topics/powersols

80. Fermat's Last Theorem: A Seventeenth Century Puzzle Solved In disgust, he wrote to Heinrich Olbers I confess indeed that fermat's theoremas an isolated proposition has little interest for me, since a multitude of http://www.wsws.org/articles/1999/jan1999/ferm-93.shtml

Enter email address to receive news about the WSWS Add Remove SEARCH WSWS English German ON THE WSWS Donate to the WSWS Expansion Fund! Editorial Board ... Books Online OTHER LANGUAGES German French Italian Russian ... Indonesian LEAFLETS Download in PDF format HIGHLIGHTS The war against Iraq and America's drive for world domination Oppose US war ... WSWS Fermat's last theorem A seventeenth century puzzle solved By Peter Symonds The following article was first published on July 23, 1993 in Workers News, the newspaper of the Socialist Labour League, the forerunner to the Socialist Equality Party (Australia). On June 23, 1993, an event took place at the Isaac Newton Institute for Mathematical Sciences at Cambridge University in Britain of considerable historic significance for the field of mathematics. In what could prove to be a major breakthrough, Andrew Wiles, a 40-year-old number theorist from Princeton University, concluded a series of three lectures on "Modular forms, elliptic curves, and Galois representations" by proving one of the longest standing problems in mathematicsFermat's last theorem, first stated around 1637. According to press reports, the proof, which is yet to be published, is up to 1,000 pages in length and uses intricate arguments from highly abstract areas of pure mathematics. Wiles' claims are yet to be checked in detail and it is possible that a flaw will be found. [Note: A major gap was found in the proof and was only resolved in October 1994. After extensive checking, the manuscripts were finally published in the May 1995 volume of the journal

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