61. Fermat's Last Theorem - An Elementary Proof By Nico De Jong (1992) Edited from the book fermat's last theorem proved by Nico de Jong (1992). http://www.geocities.com/elementaryfermat

Fermat's last theorem Was Wiles' proof really first ? Edited for the Web by Nico de Jong (c)2000 The following article is an edited version of the proof in the book Fermat's last theorem proved by Nico de Jong. Pretoria : (c)1992. ISBN 0-620-16639-8 (listed in the South African National Bibliography, Pretoria State Library, 1992, 92-2617) If you would like to comment on my proof of Fermat's last theorem, e-mail me I hope you will enjoy this article and find it worthy of discussion with your friends. PREAMBLE AND ABSTRACT After 350 years of unsuccessful attempts, a mathematically highly advanced proof of Fermat's Last Theorem (FLT) by A. Wiles was accepted and published in Annals of mathematics , May 1995. However, it cannot be Fermat's own elementary demonstration. In the opinion of the present author the following proof is the one Fermat had in mind. FLT holds that the equation z w = x w + y w can have a positive integer solution if and only if w = 2. As is well known, a proof for w being any prime suffices. Therefore w is considered a prime number throughout. Suppose z is a composite positive integer. If for only one of its prime number factors, say p

62. Is Fermat's Last Theorem Proven? An attempted elementary proof of fermat's Last theorem by James Constant, rejecting that of Wiles. http://fermat.coolissues.com/fermat.htm

63. PlanetMath: Fermat's Little Theorem fermat s Little theorem proof (Inductive) (Proof) by drini computation of powersusing fermat s little theorem (Example) by basseykay http://planetmath.org/encyclopedia/FermatsLittleTheorem.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 ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About Fermat's little theorem (Theorem) If with a prime and , then "Fermat's little theorem" is owned by KimJ view preamble View style: HTML with images page images TeX source See Also: Euler-Fermat theorem proof of Euler-Fermat theorem using Lagrange's theorem Fermat's theorem proof Other names: Fermat's Theorem Keywords: number theory Attachments: Fermat's theorem proof (Proof) by drini Fermat's Little Theorem proof (Inductive) (Proof) by drini computation of powers using Fermat's little theorem (Example) by basseykay proof of Fermat's little theorem using Lagrange's theorem (Proof) by alozano Cross-references: prime There are 6 references to this object. This is version 3 of Fermat's little theorem , born on 2001-10-15, modified 2002-03-02.

64. Proof Disproved for the same reasons fermat's Last theorem is proved by a binomial infinite series expansion http://www.coolissues.com/mathematics/Beal/beal.htm

BEAL'S CONJECTURE DISPROVED James Constant math@coolissues.com Beal's Conjecture is disproved for the same reasons Fermat's Last Theorem is proved. Beal's conjecture A prize is offered for proof or disproof of Beal's conjecture , stated as follows: If x,y,z,m,a,b positive integers then x,y,z have a common factor Proof of Fermat's Last Theorem A proof of Fermat's Last Theorem (FLT) is available using the binomial expansion . In this proof it is shown that z cannot be an integer in the equality x,y,z,m positive integers thus proving FLT x,y,z,m positive integers Disproof of Beal's Conjecture When a=b=m , Beal's equation (1) becomes Fermat's equation (2). Clearly, Fermat's equation (2) is a special case of Beal's equation (1). The same procedure used in Fermat's equation (2) can be used to show that z cannot be an integer in Beal's equation (1). Start by rearranging Beal's equation (1) x,y,z,m positive integers and then expressing the parenthesis term as a binomial series, with results 1. Since m 1/m is not an integer and the series cannot terminate becoming the binomial theorem.

65. On The Beal Conjecture An elementary proof of Beal's Conjecture given the proof of fermat's Last theorem. http://geocities.com/kerrymerry2000/

ON THE FULL BEAL CONJECTURE Consider the sum of a and b such that a 0, b p a and b are solution pairs (prior to further specification). a b p q a and b a to b m , b to a n with m n and q , all greater than 2, uniquely defining all Integer contributors. Applying T, T a b) = b m + a n p q T, once incepted, may be repeated as T z Reapplying T:, T a + b) = T( b m + a n b mn + a mn = p q which is a broad rendering of contradiction to FLT, i.e. q = mn is possible. For simplification relative to FLT, x and y are substituted to yield: x m + y m = p q q = m is potential, the expression is false (or rather an inequality) consequent to recent proof of FLT (see: WWW). Contradiction of the equivalence negates T a + b), a direct implication of T a + b). b mn + c mn unequal to p q a transformation under T:, implies b m + a n unequal to p q The object has thus been satisfied except for when p q is a power of 2, x=y=2^m, in which case q must be reduced to an equivalent 2 , which is an excepted case; and further, unless a common divisor is present, in which case factorization alters the general representation of the problem. QED Suggested reading: By same author: email: kerrymerry2000@yahoo.com

66. Fermat's Last Theorem --  Encyclopædia Britannica fermat s last theorem the statement that there are no natural numbers (1, 2, 3, elipsis;) math.x;, math.y;, and math.z; such that math.x; sup math.n http://www.britannica.com/eb/article?tocId=9034050

67. 1993: Fermat's Theorem Solved A proof of fermat s last theorem would at best provide a logical demonstration He saw a connection between the conjecture and fermat s Last theorem that http://www.capitalcentury.com/1993.html

Andrew Wiles flashes a huge grin after publicly showing off his proof for the first time in 1993. A shy and secretive Princeton University mathematics professor in 1993 unraveled a mystery that had frustrated and intrigued mathematicians for 350 years. Andrew Wiles, fascinated by math problems since age 10, figured out the last theorem of 17th century mathematician Pierre De Fermat, achieving what the most obsessed numbers crunchers of three centuries could not. The Scottish-born Wiles, in a rare interview, said the draw to solve the theorem, which stemmed from Fermat's studies of the ancient Greek text "Arithmetic," was so strong because the theorem was so simple-sounding. It says that while the square of a whole number can be broken into two other squares of whole numbers, the same cannot be done with cubes or higher powers. The theorem is based on the ancient equation developed by sixth century mathematician Pythagoreas, "X squared plus Y squared equals Z squared." The equation guided Pythagoreas' famous theory for calculating the hypotenuse of a triangle. Although Fermat himself claimed to have already proved the theorem, his notes were lost, and mathematicians, none of whom were able to solve it until Wiles, had often doubted the existence of a formal proof.

68. Fermat's Last Theorem The proof of fermat s Last theorem was completed in 1993 by Andrew Wiles, To some extent, proving fermat s theorem is like climbing Everest. http://www.cba.ua.edu/~jlee/fermat.htm

Fermat's Last Theorem Who was Fermat and what was his Last Theorem? Fermat was a 17th-century mathematician who wrote a note in the margin of his book stating a particular proposition and claiming to have proved it. His proposition was about an equation which is closely related to Pythagoras' equation . Pythagoras' equation gives you: x y z You can ask, what are the whole number solutions to this equation, and you can see that: and 5 And if you go on looking then you find more and more such solutions. Fermat then considered the cubed version of this equation: x y z He raised the question: can you find solutions to the cubed equation? He claimed that there were none . In fact, he claimed that for the general family of equations: "x n y n z n where n is bigger than 2, it is impossible to find a solution. " That's Fermat's Last Theorem. Extended Fermat's Theorem? "x n y n + w n z n where n is bigger than 2, it is impossible to find a solution." No , Naom Elkies of Harvard University discovered the following counter-example in 1988. From: Simon Singh, Fermat's Enigma, Anchor Books Inc., 1997, p. 159.

69. Fermat's Little Theorem@Everything2.com Often known as fermat s theorem, which runs the risk of confusing it with Like the similar Wilson s theorem, the proof of fermat s little theorem is not http://www.everything2.com/index.pl?node_id=682116

70. Clare College: Guide To Subjects (Fermat's Theorem) fermat s theorem is very easy to describe. Most people have come across the It is the simplicity of the statement of fermat s theorem which has made it 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

71. The Laws Of Cryptography: Fermat's Theorem Illustrated Recall that fermat s theorem says that given a prime p and a nonzero number a,ap-1 mod p is always equal to 1. Here is a table for p = 11 illustrating http://www.cs.utsa.edu/~wagner/laws/AFermat.html

The Laws of Cryptography: Fermat's Theorem Illustrated by Neal R. Wagner NOTE: This site is obsolete. See book draft (in PDF): The Laws of Cryptography with Java Code 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.)

72. Science Discussions Entry Page - Fermat's Last Theorem. fermat s last theorem a genius joke (see entry 7). Discuss your ideas onmathematics here. http://www.draaisma.net/rudi/discentries.html

SCIENCE DISCUSSION ENTRIES PAGE Mathematics (2003 - ongoing) You can react on an entry by referring to the subject of it. (Each subject has its own header color) Entry sent on Fri 06th Dec 20:04 (GMT) Subject: Fermat's last theorem Name: Ben Ito Fermat's last theorem states that there are no positive integers X. Y and Z such that, 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) Subject: Fermat 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. l. Proof.

73. 3.7 Time Delay And ``Fermat's'' Theorem 3.7 Time delay and ``fermat s theorem. gravitational lens theory in termsof timedelay and fermat s principle can be found in Blandford and Narayan 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

74. Fermat's Last Theorem: A Seventeenth Century Puzzle Solved In fact, for more than 350 years, fermat s last theorem has defeated the best Soon afterwards, DR HeathBrown proved that fermat s theorem was true for 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! Contact the WSWS ... Books Online OTHER LANGUAGES German French Italian Russian ... Indonesian LEAFLETS Download in PDF format 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

75. Andrew Wiles: Biography And Much More From Answers.com fermat s Last theorem (FLT) asserts that there are no positive integers x, y, Though less familiar than fermat s Last theorem, the TaniyamaShimura http://www.answers.com/topic/andrew-wiles

showHide_TellMeAbout2('false'); Business Entertainment Games Health ... More... On this page: Scientist Wikipedia Mentioned In Or search: - The Web - Images - News - Blogs - Shopping Andrew Wiles Scientist Wiles, Andrew [b. Cambridge, England, April 11, 1953] When he was ten years old, Wiles became interested in the solution to the problem known as Fermat's Last Theorem. Although he worked on other parts of mathematics while obtaining his doctorate at Cambridge University in 1980, he remained interested in the Fermat theorem. He began teaching mathematics at Princeton University in New Jersey in 1982. In 1986 he learned of new results that might lead to a proof of the Fermat theorem and began a secret project to use this approach to make a proof. It took seven years of intense concentration to produce a proof, but other mathematicians pointed out flaws in the 1993 publication. In collaboration with Richard L. Taylor, Wiles resolved all the difficulties and published the final proof in 1995. Wikipedia Andrew Wiles Andrew John Wiles (born April 11 ) is a British mathematician living in the United States . In , he received his bachelor's degree from the University of Oxford . He then completed his Ph.D.

76. Fermat's Theorem Using Projective Geometry Solution to fermat s theorem by CF Russell using projective geometry calculatingcircle Znuz is Znees vol. 4. Back to Contents. http://www.cfrussell.homestead.com/files/fermat.htm

Solution to Fermat's Theorem by C.F. Russell using projective geometry "calculating circle" - Znuz is Znees vol. 4. Back to Contents

77. The 2+2 Forums: Re: Was Fermat's Theorem Really Proven? Re Was fermat s theorem Really Proven? chiachu, 07/26/05 0234 AM . * Re Wasfermat s theorem Really Proven? wmspringer, 07/30/05 1239 AM http://forumserver.twoplustwo.com/showthreaded.php?Cat=&Number=3005246&page=&vie

78. The 2+2 Forums: Re: Was Fermat's Theorem Really Proven! Re Was fermat s theorem Really Proven! 07/22/05 0207 AM Was fermat s theoremReally Proven? David Sklansky, 07/22/05 0145 AM . http://forumserver.twoplustwo.com/showthreaded.php?Cat=&Number=2935521&page=&vie

79. Fermat's Last Theorem I will prove that fermat s theorem was an attempt at forming a general proof of Pythagorean s equation is also an essential part of fermat s theorem. http://physics.nad.ru/engboard/messages/562.html

Fermat's Last Theorem [ Archive message from "Physics Forum" ] Posted by Ben Ito on August 29, 2002: Fermat's Last Theorem Ben Ito I will prove that Fermat's theorem was an attempt at forming a general proof of Pythagorean's equation. l. Introduction 2. Proof The general proof of Pythagorean's equation is formed using Z^n = X^n + Y^n where n is a integer number; the lengths X, Y and Z form a right triangle only when n = 2. Rearranging Fermat's equation, Z = (X^n + Y^n)^(1/n), the lengths of X and Y form the length of Z that decreases in size when the integer value of n increases; example, n = 2, X = l m, and Y = l m, therefore, Z = (2)^(1/2) = 1.414 m; n = 50, Z = 1.01395948 m; n = 100, Z = 1.0069555 m. The largest length of Z is formed when n = 2. The value of n greater than two can never form the hypotenuse of a right triangle; consequently, n = 2 is the only solution to Fermat's equation that describes Pythagorean's right triangle. 3. Conclusion Fermat's theorem was an attempt at forming a general proof of Pythagorean's equation. Fermat stated that he could solve the problem in about a paragraph which I have done in the last paragraph of my proof. Fermat's theorem has a problem, when n = 2, the solutions are a continuous range of values; however, when n> 2 a discrete range of of solutions are described. Therefore, the whole number condition cannot be part of Fermat's theorem. Consequently, by eliminating the whole number solution condition, the general proof of Pythagorean's equation is formed using Fermat's equation.

80. NPR : Fermat's Theorem Neal talks to mathematician Keith Devlin about fermat s theorem. 17th centuryFrench mathematician Pierre fermat scribbled his enigmatic theorem on the http://www.npr.org/templates/story/story.php?storyId=1054728

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