81. Avernus - Cryptography - Fermat's Theorem This page contains a description and proof of fermat s theorem, the basis of mostsimple primality tests. http://www.avernus.org.uk/encrypt.php?article=fermat

82. The Smallest Power Congruent To 1: Fermat's Theorem The smallest power congruent to 1 fermat s theorem. 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

83. Alibris: Fermat S Last Theorem Used, new outof-print books with subject fermat s last theorem. Offering over50 million titles from thousands of booksellers worldwide. http://www.alibris.com/search/books/subject/Fermat s last theorem

You'll find it at Alibris! Log in here Over 50 million used, new and out-of-print books! CART ACCOUNT WISHLIST HELP ... SEARCH search in Books Music: All CD Vinyl Movies: All DVD VHS by title / ISBN by author / artist by subject / genre my email address unsubscribe here your shopping cart order status wish list ... help browse BOOKS Your search: Books Subject: Fermat s last theorem (19 matching titles) Narrow your results by: Hardcover Softcover Audiobook First edition ... Eligible for FREE shipping Narrow results by title Narrow results by author Narrow results by subject Narrow results by keyword Narrow results by publisher or refine further Sometimes it pays off to expand your search to view all available copies of books matching your search terms. Page of 1 sort results by Top-Selling Used Price New Price Title Author Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem more books like this by Singh, Simon Compelling, dramatic, and entirely accessible, "Fermat's Enigma" is a mesmerizing tale of heartbreak and mastery, and one that will forever change the reader's feelings about mathematics. Simon Singh co-produced an award-winning documentary film on Fermat's Last Theorem that aired on PBS's "Nova" series. 21 illustrations. see all copies from new only from first editions SVS Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem more books like this by Aczel, Amir D.

84. Mersenneforum.org - Fermat,s Theorem Incidentally my paper Euler s Generalisation of fermat s theorem a further Question relating to fermat s theorem, Acidity, Information Answers http://www.mersenneforum.org/showthread.php?t=2587

85. Mersenneforum.org - Fermat's Theorem We all know fermat s theorem that states if p is a prime and (p,a)=1 then p Question relating to fermat s theorem, Acidity, Information Answers http://www.mersenneforum.org/showthread.php?t=3543

86. Interactive Mathematics Miscellany And Puzzles Euler s theorem; fermat s theorem; Exponents of numbers; Primitive roots for primes The converse of fermat s theorem; Numbers with the fermat property http://www.cut-the-knot.org/books/ore_nums/content.shtml

Username: Password: Sites for teachers Sites for parents Awards Interactive Activities ... Sites for parents CONTENTS Preface Chapter 1. Counting and Recording of Numbers Numbers and counting Basic number groups The number systems Large numbers Finger numbers Recordings of numbers Writing of numbers Calculations Positional numeral systems Hindu-Arabic numerak Chapter 2. Properites of Numbers. Division. Number theory and numerology Multiples and divisors Division and remainders Number systems Bimu number systems Chapter 3. Euclid's Algorism Greatest conunon divisor. Euclid's algorism The division lemma Umt common multiple Greatest common divisor and least common multiple for several numbers Chapter 4. Prime Numbers Prime numbers and the prime factorization theorem Determination of prime factors Factor tables Fermat's factorization method Euler's factorization method The sieve of Eratosthenes Mersenne and Fermat primes The distribution of primes Chapter 5. The Aliquot Parts The divisors of a number Perfect numbers Amicable numbers Greatest common divisor anl least common multiple Euler's function Chapter 6. Indeterminate Problems

87. Index To Hardy And Wright 14, 18 fermat s theorem 63, 71, 85, 86, 87 fermat s Last theorem 73, 190, 202,231 fermat s theorem in k(i) 219 fermat s Problem 332 fermatEuler http://www.mathpropress.com/HardyAndWright.html

On-line Subject Index to Hardy and Wright Index for An Introduction to THE THEORY OF NUMBERS by G.H. Hardy and E.M. Wright Published by Oxford University Press, London Compiled by Robert E. Kennedy and Curtis Cooper Central Missouri State University A B C D ... Z Higher-quality versions of this index are available either as a dvi file or as a PostScript file Hardy and Wright's The Theory of Numbers INDEX Abnormal Number 21 Additive Theory of Numbers 273 Algebraic Number 159, 178 Algebraic Irrational 39 Algebraic Equation 159 Algebraic Integer 178 Algebraic Field 204 Algebraic Number 204 Almost All 8, 122 Arithmetic of Quadratic Fields 225 Arithmetical Progression 113 Arithmetical Functions 232 Associate 67, 181, 183, 305 Associate (mod m) 89 Asymptotically Equivalent 8 Average Order 263, 272 Bachet's Weights Problem 115 Bauer's Identical Congruence 98, 100, 102 Belongs to 71 Bernoulli's Numbers 90 Bertrand's Postulate 343 Big-Oh Notation 7 Binary Decimal 111 Binomial Coefficient 63 Biquadrate 317, 327 Bohr's Proof 388 Boundary 31 Bounded Quotients 165

88. AoPS Math Forum :: View Topic - Darboux Theorem Post Posted Mon Feb 23, 2004 904 pm Post subject fermat s theorem Can someone tell me what fermat s theorem say (maybe we have the same in french in http://www.artofproblemsolving.com/Forum/topic-4125.html

Round 1 of the USA Mathematical Talent Search is now available. The postmark deadline for the first round is October 3 Font Size: The time now is Fri Sep 16, 2005 4:28 pm All times are GMT - 7 Hours Darboux Theorem Mark the topic unread Math Forum College Playground ... Theorems, Formulas and Other Questions Moderators: blahblahblah fedja Kent Merryfield Moubinool ... Myth Page of [6 Posts] Printer View Tell a Friend View previous topic View next topic Author Message Moubinool Navier-Stokes Equations Joined: 27 Aug 2003 Posts: 1979 Posted: Fri Feb 20, 2004 4:11 pm Post subject: Darboux Theorem Darboux Theorem states that the derivative of a function has the Intermediate Value Property Darboux property. thank you Lagrangia Back to top New Member Joined: 11 Jan 2004 Posts: 13 Posted: Sun Feb 22, 2004 5:54 am Post subject: thanks a lot, moubi Back to top liyi Navier-Stokes Equations Joined: 17 Jul 2003 Posts: 1534 Posted: Sun Feb 22, 2004 7:06 am Post subject: Re: Darboux Theorem Moubinool wrote: Back to top liyi Navier-Stokes Equations Joined: 17 Jul 2003 Posts: 1534 Posted: Mon Feb 23, 2004 8:54 pm

89. Euler's Theorem And Small Fermat's Theorem This article is concerned with Euler s theorem and small fermat s theorem thatplay important roles in publickey cryptograms. In the first section, http://mizar.uwb.edu.pl/JFM/Vol10/euler_2.html

Journal of Formalized Mathematics Volume 10, 1998 University of Bialystok Association of Mizar Users Euler's Theorem and Small Fermat's Theorem Yoshinori Fujisawa Shinshu University, Nagano Yasushi Fuwa Shinshu University, Nagano Hidetaka Shimizu Information Technology Research Institute, of Nagano Prefecture Summary. MML Identifier: The terminology and notation used in this paper have been introduced in the following articles [ Contents (PDF format) Preliminary Finite Sequence of Naturals Modulus for Finite Sequence of Naturals Euler's Theorem and Small Fermat's Theorem Acknowledgments The authors wish to thank Professor A. Trybulec for all of his advice on this article. Bibliography 1] Grzegorz Bancerek. Cardinal numbers Journal of Formalized Mathematics 2] Grzegorz Bancerek. The fundamental properties of natural numbers Journal of Formalized Mathematics 3] Grzegorz Bancerek. Joining of decorated trees Journal of Formalized Mathematics 4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences Journal of Formalized Mathematics 5] Czeslaw Bylinski. Functions and their basic properties Journal of Formalized Mathematics 6] Czeslaw Bylinski.

90. Fermat fermat s theorem states that if p is a prime and a is not a multiple of See his paper, Tests for primality by the converse of fermat s theorem Bull. http://www.dozenalsociety.org.uk/numbers/fermat.html

Notes on the converse of Fermat's theorem Composite with factors unknown Some students are puzzled on reading that such and such a (large) number is composite but its factors are not known. Books in which such statements appear - and there are quite a few of them in the 'popular' mathematics field - should include a few lines to explain how this can be true. Fermat's theorem states that if p is a prime and a is not a multiple of p, then (a p-1 -1) is a multiple of p. Or in congruence language a p-1 = 1 (mod p). Though this is true we may not infer that the converse is necessarily true, namely that if a n-1 = 1 (mod n) then n must be prime. This n may very well be prime, but there are many composite numbers which satisfy the congruence. Lucas first laid down the necessary conditions for a true converse of the theorem, namely that "If a x =1 (mod n) for x = n-1, but not for x a proper divisor of n-1, then n is a prime." For example, eleven is a prime, and 2 = 1 mod(11); x here is 10, and there is no proper divisor, d, of 10 for which 2 d = 1 mod 11.

91. The Mathematics Of Fermat's Last Theorem Paulo Ribenboim fermat s Last theorem for Amateurs Ribenboim has written Alf van der Poorten - Notes on fermat s Last theorem Van der Poorten s book http://cgd.best.vwh.net/home/flt/flt01.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 enjoy mathematics, you should find the effort very rewarding. Please let me know by email how you like these pages. Let me know if you find any errors or passages which can readily be improved (short of duplicating what's in the references listed below). Good news! Many people have asked whether the following pages of this site are available in a printable or other offline format. Apollo Hogan has generously provided TeX versions of the pages here (as of November 1997). The TeX has been processed into both DVI and PostScript forms for viewing and printing. Select from the following links to download the document in the format you prefer. (Note: These files are not as recent as the site itself, though as of this writing only minor editorial changes have been made to the site.) PDF files This is a ZIP file containing separate PDF files for each section of the site. Adobe Acrobat is required to view the files, and an un-ZIP utility is required to remove them from the ZIP archive. Size: 170KB.

92. The Mathematics Of Fermat's Last Theorem The statement of fermat s Last theorem (FLT for short) is about as simple as any MacTutor History of fermat s Last theorem Good introduction with many http://cgd.best.vwh.net/home/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.

93. EULER'S FUNCTION; EULER'S THEOREM; FERMAT'S THEOREM EULER S FUNCTION; EULER S theorem; fermat S theorem. http://www.ece.utexas.edu/~takis/teaching_material/infocrypto/Numbers/htmlfiles/

Next: MULTIPLICATIVE FUNCTIONS Up: Incongruences Previous: SETS OF RESIDUES Contents 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 If then The special case where , a prime, is called Fermat's theorem If is prime then, for all 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 be the number of elements of . Obviously, . Now define an equivalence relation: iff for some integer . Let be the equivalence class of under this relation. It is easy to see that has exactly the same number of elements as , i.e. . Hence is a multiple of . Since , it follows that as well. We study the structure of Euler's function further. Recall that is the set of equivalence ( ) classes of integers relatively prime to . Alternatively, by picking a representative from each class, in a way that this representative is between and , we think of as a so-called reduced set of residues . If , there is a natural one-to-one correspondence between and . This is defined by the function First we see that is indeed in . But . Since , it follows that . So . Similarly

94. Fermat Corner Welcome to the fermat Corner. FInd out more about fermat s Last theorem, AndrewWiles 1 bestseller fermat s Last theorem (aka fermat s Enigma in the US) http://www.simonsingh.com/Fermat_Corner.html

Fermat Corner Back to Homepage The Whole Story Who was Fermat? What is the Theorem? ... Wolfskehl Prize Andrew Wiles Fermat Corner Fermat’s Last Theorem is the most notorious problem in the history of mathematics and surrounding it is one of the greatest stories imaginable. This section explains what the theorem is, who invented it and who eventually proved it . When finished, it will also tell the fascinating stories of the some of the other mathematicians whose lives were tormented by this beautiful and intriguing problem. Fermat’s 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 Fermat’s 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 Fermat’s Last Theorem taught me to love mathematics for its own sake. There is a Mathematics Corner currently being developed for this site.

95. FERMAT'S LAST THEOREM By SIMON SINGH. Buy At Www.thebookplace.co.uk fermat S LAST theorem by SIMON SINGH. This work tells the true story of howfermat s theorem was made to yield up its secrets. http://search.bookplace-ltd.co.uk/book_info/0001054635_FERMAT'S-LAST-THEOREM_SIM

FERMAT'S LAST THEOREM - SIMON SINGH Andrew Wiles had dreamed of proving Fermat ever since he first read about the theorem as a boy of ten at his local library. Only after years of toil, frustration and disappointment came the breakthrough. This work tells the true story of how Fermat's Theorem was made to yield up its secrets. SEARCH BROWSE SUBJECTS HELP FERMAT'S LAST THEOREM SIMON SINGH Andrew Wiles had dreamed of proving Fermat ever since he first read about the theorem as a boy of ten at his local library. Only after years of toil, frustration and disappointment came the breakthrough. This work tells the true story of how Fermat's Theorem was made to yield up its secrets. Audiobook More information. Buy. BP BE

96. Fermat S Last Theorem - Information Technology Services Wiles did not even look at fermat s last theorem. He proved a theorem about Frey realized that if p,q,r, satisfied fermat s theorem, then a simple http://www.physicsforums.com/archive/t-63382_Fermat's_last_theorem.html

Technology Services Mathematics Number Theory Fermat's last theorem - Fermat's last theorem Hi everybody, I wonder what knowledge is required to study Wiles proof of the last theorem of Fermat. Of course, i don't mean fully understand it but just get to understand some of his thoughts and how he actually approached the problem and found the solution. I would really like to hear the experience of anyone that has studied it for a while. Just make something clear: I wouldn't like very complicated explanations as I am really new in number theory. I have until now studied mostly analysis and linear algebra but i am now finding number theory a really interesting field. Thanks Discuss Fermat's last theorem Here, Free! arildno - Fermat's last theorem Well, you should start with Simon Singh's "Fermat's last theorem" book. Discuss Fermat's last theorem Here, Free! mathwonk - Fermat's last theorem I have not read Wiles paper but have read Singh's book, and I recommend it highly. I know a little about the proof though, as follows. Wiles did not even look at Fermat's last theorem. He proved a theorem about elliptic curves, which had been conjectured by Frey, and proved by Ribet, to imply Fermat's theorem.

97. Fermat S Theorem Math Problem - Information Technology Services Discuss fermat s theorem math problem Here, Free! Become A Member, Free!robert Ihnot fermat s theorem math problem. x1Mod 5, Y1 Mod 5, Z0 Mod 5. http://www.physicsforums.com/archive/t-62774_Fermat's_theorem_math_problem.html

Technology Services Mathematics General Math Fermat's theorem math problem - Fermat's theorem math problem from: http://www.math.utah.edu/~cherk/puzzles.html Fermat, computers, and a smart boy A computer scientist claims that he proved somehow that the Fermat theorem is correct for the following 3 numbers: x=2233445566, y=7788990011, z=9988776655 He announces these 3 numbers and calls for a press conference where he is going to present the value of N (to show that x^N + y^N = z^N and that the guy from Princeton was wrong). As the press conference starts, a 10-years old boy raises his hand and says that the respectable scientist has made a mistake and the Fermat theorem cannot hold for those 3 numbers. The scientist checks his computer calculations and finds a bug. How did the boy figure out that the scientist was wrong? I am stumped, I noticed the pattern in the digits of the numbers, but I do not see how I can link that to the possibility of forming such a statement with those numbers when n is greater than 2. Discuss Fermat's theorem math problem Here, Free!

98. Countrybookshop.co.uk - Fermat's Last Theorem Andrew Wiles had dreamed of proving fermat ever since he first read about thetheorem as a boy of ten at his local library. Only after years of toil, http://www.countrybookshop.co.uk/books/index.phtml?whatfor=0001054635

99. Faces Of The Moon: Pinoy's Feat On Fermat's Theorem Turns Out To Be Fake Other mathematicians attempted to prove fermat s Last theorem, but unfortunatelyto no avail. And this theorem went unproven for more than 300 years. http://perryv.blogspot.com/2005/05/pinoys-feat-on-fermats-theorem-turns.html

@import url("http://www.blogger.com/css/blog_controls.css"); @import url("http://www.blogger.com/dyn-css/authorization.css?blogID=10871028"); @import url(http://www.blogger.com/css/navbar/main.css); @import url(http://www.blogger.com/css/navbar/1.css); BlogThis! Faces of the Moon Temporary site only, while the real one is down. The real one is http://perryv.i.ph Tuesday, May 24, 2005 Pinoy's feat on Fermat's Theorem turns out to be fake Yep, I concede, I was also a victim of this one. Please forgive me for the misinformation. :-) Maybe I trusted Manila Times too much, thinking that it was a reliable source. My fault too for the lack of research. :-( But thanks very much to Alecks Pabico's enlightening research. He himself wrote to Prof. Andrew Wiles about the alleged letters to Prof. Edgar Escultura. Wiles denied that the letters came from him. See more of Alecks' findings here. Thank you very much Alecks for your investigation. :-) In 1637, the French mathematician Pierre De Fermat stated that, for any nonzero counting numbers x, y, z and n, there are no numbers x, y, and z

100. Math 696 -- Public-key Cryptography The basis of the method is fermat s theorem from number theory. According toEuler s generalization of fermat s little theorem, if a and n are positive http://www.math.tamu.edu/~harold.boas/courses/math696/public-key-cryptography.ht

Public-key cryptography Public-key cryptography With the growth of electronic communications, privacy has become a significant issue on the Internet. There is a newsgroup alt.privacy , a vast literature on anonymous email , and services like The Anonymizer and IDzap that offer anonymous World-Wide Web surfing. Average computer users would likely be satisfied with the capability of putting their email in an envelope. In the ordinary course of events, email is more like a postcard: it can easily be read in passing. Private email is now widely and freely available through public-key cryptography To use classical ciphers, the sender and recipient must exchange a common key ahead of time. The idea behind public-key cryptography is that one key can be used to encode a message, and a different key can be used to decode the message. A user can make public one of a pair of complementary keys to enable the world to send enciphered messages that can be deciphered only by that user. Also, the user can broadcast enciphered messages that everyone can decipher and verify as coming from that user. Thus, public-key cryptography simultaneously meets the goals of privacy and of authentication. The first mathematical implementation of a public-key cryptosystem was published by R. L. Rivest

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