61. 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.org/blue/Fermat.shtml

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Fermat's Little Theorem 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.

62. Fermat's Last Theorem And The Fourth Dimension fermat s Last theorem and the Fourth Dimension. The rst problem has no solution;in fact, this is just the case n = 3 of fermat s Last theorem. http://www.g4g4.com/4d/4djimp.htm

home g4g4.com Fermat's Last Theorem and the Fourth Dimension Jim Propp: http://www.math.wisc.edu/ ù propp/flt4d.html. page 1 their wits on. He thought that these challenges would give others a greater appreciation of the hidden depths surrounding his problems about numbers and lure them into doing active research on the topic, but sometimes the tactic back red on him. For instance, in one of his letters he challenged the English mathematician John Wallis to solve two problems: 1. given a cube,to write that cube as a sum of two cubes; and 2. given a sum of two cubes, to write that number as a sum of two cubes in a dierent way. page 2 page 3 page 4 For pages 5 - 14, please open pdf file. Click Here For more information, please visit: http://www.math.wisc.edu/ ù propp/flt4d.html.

63. 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

Fermat's Theorem. Congruent Modulo. If p is prime and a is a positive integer not divisible by p then: a (p-1) is equivalent to a p Let's find out the mod of a divisor with a large exponent. However, first let's use an exponent that isn't extremely large and then afterwards use a large one. ) mod(13) )] mod(13) ) mod(13)] * [(6 mod(13)] ) 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 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 . So, according the Fermat's theorem a (p - 1) mod p = 1, we have (6 ) mod(13) = 1, which leaves something easily calculable 6 mod(13) = 2. Let's now look at a very large exponent: ) mod(7) )] mod(7) ) mod(7)] * [(5 ) mod(7)] ) mod(7) = 3 The only difference between this problem and the one above is that 1998 is a multiple of p - 1, hence 6 * 333 = 1998. Now what happens when the dividend is very large (or larger than the exponent)? The answer lies within the Chinese Remainder Theorem. Fermat's Last Theorem.

64. Fermat's Theorem fermat s theorem. The 17th century mathematician Pierrede fermat produced many theorems. This could mean http://www.fact-index.com/f/fe/fermat_s_theorem.html

Main Page See live article Alphabetical index Fermat's theorem The 17th century mathematician Pierre de Fermat produced many theorems. This could mean: Fermat's last theorem Fermat's little theorem others? This is a disambiguation page This article is from Wikipedia . All text is available under the terms of the GNU Free Documentation License

65. Proofs Of Fermat's Little Theorem Proofs of fermat s little theorem. This is a collection of proofs of fermat s littletheorem a p = a (mod p) for every prime number p and every integer a. http://www.fact-index.com/p/pr/proofs_of_fermat_s_little_theorem.html

Main Page See live article Alphabetical index Proofs of Fermat's little theorem This is a collection of proofs of Fermat's little theorem a p a mod p for every prime number p and every integer a Note that it is enough to prove a p = 1 (mod p for every integer a which is relatively prime to p (i.e. not a multiple of p ). Multiplying with a then gives the above version of the theorem for those numbers a ; for the multiples of p the above version is clear anyway. Table of contents 1 A direct proof 2 Inductive proof with the binomial theorem 3 A proof using bracelets 4 A proof using dynamical systems ... 6 A proof using group theory A direct proof We will assume a to be relatively prime to p . This proof will make use of our base a multiplied by all the numbers from 1 to p -1. It turns out that if p is prime, the values 1 a through ( p a (modulo p ) are just the numbers from 1 through p -1 rearranged, a consequence of the following lemma. We then multiply all those numbers together, resulting in a formula from which the theorem follows. Lemma: If a is relatively prime to p and x and y are integers such that xa ya (mod p ), then

66. Proving Fermat's Little Theorem By Induction fermat s Little theorem. Proving fermat s Little theorem By Induction.fermat s Little theorem a p =amodp. Proof Consider 1 p =1modp. http://www.users.globalnet.co.uk/~perry/maths/fermatbyinduction/fermat.htm

Fermat's Little Theorem [Home] [Other Maths] Proving Fermat's Little Theorem By Induction Fermat's Little Theorem a p =amodp Proof Consider 1 p And create the inductive relationship: (a+1) p =[a p p ]modp For a=1, we have 2 p p p ]modp=2modp Then a=2 gives 3 p p p ]modp=3modp and so on. This fails to work for all n composite because the binomial expansion of (a+1) n does not have inner terms that always divide n. Please address questions/comments/suggestions to : Jon Perry

67. PlanetMath: Fermat's Theorem (stationary Points) fermat s theorem (stationary points), (theorem). Let Attachments proofof fermat s theorem (stationary points) (Proof) by paolini. Cross http://planetmath.org/encyclopedia/FermatsTheoremStationaryPoints.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... Random Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Corrections talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information Docs Classification News Legalese ... TODO List Fermat's Theorem (stationary points) (Theorem) Let be a continuous function and suppose that is a local extremum of . If is differentiable in then "Fermat's Theorem (stationary points)" is owned by paolini view preamble View style: HTML with images page images TeX source Attachments: proof of Fermat's Theorem (stationary points) (Proof) by paolini Cross-references: differentiable extremum continuous function There is 1 reference to this object. This is version 2 of Fermat's Theorem (stationary points) , born on 2003-07-15, modified 2003-07-15. Object id is 4450, canonical name is FermatsTheoremStationaryPoints. Accessed 392 times total. Classification: AMS MSC (Real functions :: Functions of one variable :: One-variable calculus) Pending Errata and Addenda None.

68. Fermat S Last Theorem - The Story, The History And The Mystery fermat s Last theorem. David Shay. The birth of the problem Proofs forspecial cases First steps in general proofs The computer enters http://fermat.workjoke.com/

Fermat's Last Theorem David Shay The birth of the problem Proofs for special cases First steps in general proofs The computer enters the picture ... David Shay David's Homepage Profession Jokes - Mathematicians

69. Fermat S Last Theorem fermat s Last theorem. The high point of Greek number theory was thedetermination of all Pythagorean triples by Euclid 15, Book http://math.nmsu.edu/~history/monthly/node4.html

Next: The parallel postulate Up: Great Problems of Mathematics: Previous: Solutions of algebraic Fermat's Last Theorem The high point of Greek number theory was the determination of all Pythagorean triples by Euclid [ , Book X, Lemmas 1,2; in v. 3, p. 63f,] and Diophantus. The motivation was of course geometric, namely to determine all right triangles with integer sides, via the Pythagorean Theorem. Diophantus' Arithmetica ] inspired Fermat to conjecture in the margin of his copy what is now known as Fermat's Last Theorem, arguably the most famous open problem in all of mathematics. (Fermat's annotation can be found in [ , p. 2,] [ , p. 218,] [ , p. 213,].) Fermat probably could prove the conjecture for , but it was left to Euler to publish the first explicit proofs (which contained a gap for n = 3 ). Euler's proof for n = 4 , pp. 3637,] is quite accessible, using Fermat's method of ``infinite descent'' to reduce the problem to the determination of Pythagorean triples. (See e.g. [ , pp. 57,] for a rigorous classification of all Pythagorean triples.)

70. Fermat's Last Theorem Article on fermat s last theorem from WorldHistory.com, licensed fromWikipedia, the free encyclopedia. Return Index fermat s last theorem. http://www.worldhistory.com/wiki/F/Fermat's-last-theorem.htm

World History (home) Encyclopedia Index Localities Companies Surnames ... This Week in History Fermat's last theorem Fermat's last theorem in the news Fermat's last theorem (also called Fermat's great theorem ) is one of the most famous theorem s in the history of mathematics . It states that: There are no positive natural numbers a b , and c such that a^n + b^n = c^n in which n is a natural number greater than 2. 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 but this margin is too small to contain it". 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 last to be proved . The theorem is generally thought to be the mathematical result that has provoked the largest number of incorrect proofs. For various special exponents n , the theorem had been proved over the years, but the general case remained elusive. In

71. Euler's Theorem And Small Fermat's Theorem Euler s theorem and Small fermat s theorem. Finally, Euler s theoremand small fermat s theorem are proved. MML Identifier EULER_2. 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.

72. Avernus - Cryptography - Fermat's Theorem This page contains a description and proof of fermat s theorem, thebasis of most simple primality tests. fermat s theorem. Although http://www.avernus.org.uk/encrypt.php?article=fermat

73. Clare College: Guide To Subjects (Fermat's Theorem) fermat s theorem. It describes, in language intended for the intelligent layman ,the history, the importance and the proof of fermat s theorem.). http://www.clare.cam.ac.uk/admissions/subjects/fermat.html

CLARE COLLEGE Admissions Home About Clare ... Search Fermat's Theorem (The article below was written for the Clare Association Annual in 1994. It describes, in language intended for the "intelligent layman", the history, the importance and the proof of Fermat's Theorem.) Last year saw possibly the most remarkable developments in mathematics this century - and Clare College was at the centre of it. For over 350 years mathematicians all over the world have tried without success to prove what has become known as ``Fermat's Last Theorem''. But recently a proof was announced by Andrew Wiles, using the help of Richard Taylor. A.J. Wiles (1974) was a graduate student, and then Research Fellow, in the late 1970s and is now a Professor at Princeton University; R.L. Taylor (1980), who was Wiles' research student, is currently a Fellow of Clare and Reader in Pure Mathematics. Does a n +b n =c n have a solution in integers if and Fermat's theorem is very easy to describe. Most people have come across the fact that 3 (most likely during a study of right-angled triangles). Similar examples, such as 5

74. Chike Obi And Fermat's Last Theorem mathematician, Prof. Chike Obi, has given scientific proof to a 361yearold mathematical puzzle known as fermat s Last theorem. The theorem http://www.math.buffalo.edu/mad/PEEPS/obi-chike-fermat.html

re-printed from a Nigerian news article: Chike Obi solves 361-year-old maths puzzle. By Akin Jimoh BY plain brainwork and without the use of modern technological aid such as computers, world acclaimed Nigerian mathematician, Prof. Chike Obi, has given scientific proof to a 361-year old mathematical puzzle known as Fermat's Last Theorem. The theorem, well known among mathematicians and other allied professions, was enunciated by one of the two leading mathematicians of the first half of the 17th Century, Pierre de Fermat, a French. By far, the best known of Fermat's many theorems, it states that the equation xn+yn=zn; where x,y,z, and n are positive integers, has no solution if n is greater than two. Fermat had, on this particular theorem, which appeared in the margin of Diophantus Arithmetic, stated: "I have discovered a truly wonderful proof of this proposition, but the margin is too small to contain it." Besides, measures will shortly be put in place by the academy to popularise Obi's works and other breakthroughs in science in Nigeria. Another professor of mathematics, Jerome Ajayi Adepoju who described Obi as "my senior colleague for many years," said the simplicity of the method employed by the University of Lagos emeritus professor stands it out from what other mathematicians have done. This, according to him, proved that Fermat, in his simplistic way and using the technology of the 17th Century, was indeed right.

75. Fermat's Last Theorem fermat s Last theorem. from moviemistakes.com. Some trivia about theFaustian film Bedazzled (which came out in the year 2000 and http://www.pen.k12.va.us/Div/Winchester/jhhs/math/facts/fermat.html

Fermat's Last Theorem from moviemistakes.com Some trivia about the Faustian film "Bedazzled" (which came out in the year 2000 and stars Elizabeth Hurley and Brendan Frazier): "This is not a mistake but something clever. When the Devil was a schoolteacher, the first equation on the board was Fermat's Last Theorem (X to the Nth + Y to the Nth = Z to the Nth for N is greater than 2). It was proven in 1993 that there is no integer solution to this equation; exactly the kind of problem the Devil would give to an algebra class, but not really expected in your standard Hollywood film." Links: Math Facts Page Handley Math Home Page

76. Fermat's Last Theorem - Wikipedia, The Free Encyclopedia fermat s last theorem. fermat s last theorem (also called fermat s Great theorem)states that David Shay fermat s last theorem, http//fermat.workjoke.com/. http://www.phatnav.com/wiki/wiki.phtml?title=Fermat's_last_theorem

77. Fermat's Last Theorem -- From Eric Weisstein's Encyclopedia Of Scientific Books fermat s Last theorem. see also fermat s Last theorem. Aczel, Amir D. fermat sLast theorem Unlocking the Secrets of an Ancient Mathematical Problem. http://www.ericweisstein.com/encyclopedias/books/FermatsLastTheorem.html

Fermat's Last Theorem see also Fermat's Last Theorem Aczel, Amir D. Fermat's Last Theorem: Unlocking the Secrets of an Ancient Mathematical Problem. New York: Four Walls Eight Windows, 1996. 147 p. $18. Bell, Eric Temple. The Last Problem. Washington, DC: Math. Assoc. Amer., 1990. 326 p. $18.50. Edwards, Harold M. Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory. New York: Springer-Verlag, 1977. 410 p. $59. Murty, V. Kumar (Ed.). Seminar on Fermat's Last Theorem. Providence, RI: Amer. Math. Soc., 1995. 265 p. $49. Osserman, Robert (Ed.). Fermat's Last Theorem. The Theorem and Its Proof: An Exploration of Issues and Ideas. 98 min. videotape and 56 p. book. 1994. $29.95. Ribenboim, Paulo. 13 Lectures on Fermat's Last Theorem. New York: Springer-Verlag, 1979. 302 p. Nice book. Chock full of information in usual Ribenboim style. $59.95. Ribenboim, Paulo. Fermat's Last Theorem for Amateurs. New York: Springer-Verlag, 1999. 424 p. $39.95. Singh, Simon. Fermat's Enigma: The Quest to Solve the World's Greatest Mathematical Problem. New York: Walker and Co., 1997. 288 p. $23. Stewart, Ian and Tall, David.

78. Powell's Books - Fermat's Last Theorem: Unlocking The Secret Of An Ancient Mathe From formulas devised for the farmers of ancient Babylonia to the dramatic proofof fermat s theorem in 1993, this extraordinary work takes us along on an http://www.powells.com/cgi-bin/biblio?inkey=2-0385319460-1

79. 3.7 Time Delay And ``Fermat's'' Theorem 4 Lensing Phenomena 3 Basics of Gravitational Lensing 3.6 Lens mapping.3.7 Time delay and ``fermat s theorem. The deflection angle http://relativity.livingreviews.org/Articles/lrr-1998-12/node10.html

3.7 Time delay and ``Fermat's'' theorem The deflection angle is the gradient of an effective lensing potential (as was first shown by [ ]; see also [ ]). Hence the lens equation can be rewritten as or The term in brackets appears as well in the physical time delay function for gravitationally lensed images: This time delay surface is a function of the image geometry ( ), the gravitational potential , and the distances , and . The first part - the geometrical time delay - reflects the extra path length compared to the direct line between observer and source. The second part - the gravitational time delay - is the retardation due to gravitational potential of the lensing mass (known and confirmed as Shapiro delay in the solar system). From Equations ( ), it follows that the gravitationally lensed images appear at locations that correspond to extrema in the light travel time, which reflects Fermat's principle in gravitational-lensing optics. The (angular-diameter) distances that appear in Equation ( ) depend on the value of the Hubble constant [ ]; therefore, it is possible to determine the latter by measuring the time delay between different images and using a good model for the effective gravitational potential

80. The Smallest Power Congruent To 1: Fermat's Theorem The smallest power congruent to 1 fermat s theorem. fermat s theoremFor any number a relatively prime to the prime p, we have. 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

Page 4     61-80 of 111    Back | 1  | 2  | 3  | 4  | 5  | 6  | Next 20