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1. Fermat's Last Theorem -- From MathWorld
Fermat s Last Theorem. Bell, E. T. The Last Problem. New York Simon and Schuster,1961. Cipra, B. A. fermat theorem Proved. Science 239, 1373, 1988.
http://mathworld.wolfram.com/FermatsLastTheorem.html

2. Untitled Document
An attempted elementary proof.
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3. Little Fermat Theorem
Little fermat theorem. (2)(2 10 ) n 1 is divisible by 341. X (p-1) - Y (p-1)= 0 mod(p) is another interesting fact proven by the Little fermat theorem.
##### Little Fermat Theorem
The Little Fermat Theorem states that if p is a prime then p divides a p -a, p not equal to a. After factoring : a(a (p-1)
The converse of the theorem is not true, in fact there are composite numbers that divide a p -a. They are referred to as psuedoprimes The Fermat numbers denoted by: F n are always primes or psuedoprimes when a=2.
n +1 is the suspected prime to be tested.
Let a = 2 then: 2 n -2, or 2[2 n -1] is divisible by 2 n
Let f = 2 n , This is a number that only has 2 as its factor.
That leaves us with the expression 2 f -1 to factor.
Using the fact that x -y factors to (x+y)(x-y)
then 2 f -1 factors to (2 (f / 2) (f / 2)
Factoring the -1 factor repeatedly until the exponent is 1 we get:
f -1 factors to (2 (f / 2) (f / 4) (f / 8) (f / 16) n
Example: +1 is F , or 2 +1 and 2 Substituting all values in the original expression we get 2(2 -1) is divisible by Factoring: 2 Fermat , the father of number theory, conjectured that all 2 n +1 were prime. the first four are but 2 +1 is composite. Euler proved that this number is divisible by 641. (more on all of this later) I have played with this expression trying to find a pattern to psuedoprimes . I have found that if a has the pattern of kp+1 or (kp+1) t then any number p divides the expression: a n -a.

4. Little Fermat Theorem
(X (p1) -1) - (Y (p-1) -1) = m, Associate and factor a (-1). p(t) - p(u) = m,Factor p via Little fermat theorem. p(tu) = pk, Substituting m=pk and Factoringp.
##### X (p-1) - Y (p-1) = mod(p)
X (p-1) - Y (p-1) = m X and Y not equal to p, a prime X (p-1) (-1) - Y (p-1) (+1) =m Subtract and Add a 1 (X (p-1) -1) - (Y (p-1) -1) = m Associate and factor a (-1) p(t) - p(u) = m Factor p via Little Fermat Theorem p(t-u) = pk Substituting m=pk and Factoring p X (p-1) - Y (p-1) = mod(p) Substituting and applying mod

5. Euler-Fermat Theorem
First of all, Fermat s theorem states that if p is a prime, then a = a p mod(p)where a is any integer If we divide by a on both sides of this equation we get
 First of all, Fermat's theorem states that if p is a prime , then a = a p mod(p) where a is any integer If we divide by a on both sides of this equation we get: 1 = a p-1 mod(p) = a j (p) mod(p) by the Euler-phi function Euler then expands this theorem to state that if gcd(m,n) = 1 then, m j (n) mod(n) = 1 Using this theorem, we can show that if gcd(m,n) = 1 and we encrypt a message, m with the formula, e=m k then we will always get the same m back if we decrypt it with the formula, m=e d Proof: e = m k mod(n) #1: e d mod(n) = m kd mod(n) From the back-substitution done in Euclid's algorithm we have: (i)(k) = (j)( j (n)) + 1 If i was positive then we have: d = i so (d)(k) = (j)( j (n)) + 1 We can simply take b = j, whatever number j (n) is multiplied by. If i was negative then we have: d = j (n) + i so i = d - j (n) therefore, (i)(k) = (j)( j (n)) + 1 (d - j (n))(k) = (j)( j (n)) + 1 (d)(k) - ( j (n))(k) = (j)( j (n)) + 1 (d)(k) = (j)( j (n)) + 1 + ( j (n))(k) (d)(k) = (j)( j (n)) + ( j (n))(k) + 1 (d)(k) = (j + k)( j (n)) + 1 Once again, we can take b = (j + k), whatever number

6. PlanetMath: Euler-Fermat Theorem
Eulerfermat theorem, (Theorem). Given , when gcd , where is the Euler totientfunction. See Also Fermat s little theorem, Fermat s theorem proof
http://planetmath.org/encyclopedia/EulerFermatTheorem.html

7. The Last Fermat Theorem
The Last fermat theorem. The following paragraphs contain a shortoutlook on the Last fermat theorem with regards to Gsystems.
##### The Last Fermat theorem
The following paragraphs contain a short outlook on the Last Fermat theorem with regards to G-systems.
##### Instances in segments
Let c(s) be a number of instances in segment s. In case a^p +b^p =c^p (i.e. the Last Fermat theorem) it should hold: For example in G(p) the values c(s) are: p=2: 1, 3, 5, 7, 9, 11 13, 15, 17, 19, 21, 23, 25, 27 .. p=3: 1, 7, 19, 37, 61, 91, 127, 169, 217, 271, ... p=5: 1, 31, 211, 781, 2101, 4651, ... p=7: 1, 127, 2059, 14197, ... Only in case p=2 such sums are known:
In case p=3, i.e. G(3) the values c(s) are: In the table it holds: (2a) R[i,j]=R[i,1]+R[i+1,j-1] E.g. R[3,3]=R[3,1]+R[4,2]=19+98=117 Written in an other way: (2b) R[i,j]=R[i-1,j+1]-R[i-1,1] E.g. R[4,2]=R[3,3]-R[3,1]=117-19=98
##### Congruences
The equation (1) must hold also for every module m: Therefore, sum of numbers in a block of some adjacent rows from the first table should be equal to the numbers of one row from the second table (for each column).
##### Some small dependencies
From the expression (b+p)^p-b^p = (mod p) and from the structure of diferential progressions of self-classes follows: s[(b+p)^p-(b+p)]/p - (b^p-b)/p = -1 (mod p ) (b+p)^p - b^p = ( mod p^2 ).

8. G.Imbalzano On II Fermat'theorem & Structure Fine'constant.
News 2003 for SSGRR Universe for Life ! From Holy Shroud to BigBang ! ~~~Use MS LineDraw (fixed) FONT~~~ Associazione Insegnanti
http://users.libero.it/jmbalzan/newsiol.htm
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9. Mbox: Wiles' Proof Of The Fermat Theorem
Wiles proof of the fermat theorem. Zdzislaw Meglicki (Zdzislaw.Meglicki@cisr.anu.edu.au)Mon, 14 Nov 1994 160644 +1100 (EST)
http://www-unix.mcs.anl.gov/qed/mail-archive/volume-2/0077.html
##### Wiles' proof of the Fermat Theorem
Zdzislaw Meglicki Zdzislaw.Meglicki@cisr.anu.edu.au
Mon, 14 Nov 1994 16:06:44 +1100 (EST)
In the last issue of the New Scientist, I've found a brief note that
Andrew Wiles has fixed the problem in his proof of the last Fermat
Theorem, which should really be renamed to "Fermat-Wiles" theorem, if
the proof is correct. Chatting about it with John Slaney, we came to the
conclusion that the verification of that proof would be an ideal Holy
Grail for QED. In other words, if you could use the QED system in order
to verify a proof as complex and convoluted as Wiles' proof, you'd
demonstrate to all mathematicians enormous usefulness of such a system.
Greetings to all,
Zdzislaw Meglicki, Zdzislaw.Meglicki@cisr.anu.edu.au The Australian National University, Canberra, A.C.T., 0200, Australia, fax: +61-6-249-0747, tel: +61-6-249-0158

10. Fermat's Little Theorem
Another private reply John, In Carmichael s 1914 number theory text, he says it is often referred to as the simple fermat theorem. He also attributes
http://www.spd.dcu.ie/johnbcos/fermat's_little_theorem.htm
 Fermat's little theorem Michel Waldschmidt on Fermat Friday 17 th August 2001 was the 400 th anniversary of the birth of Pierre de Fermat , and by way of a personal homage I decided that for the ICTMT5 meeting in Klagenfurt, Austria - held the week prior to Fermat's anniversary - I would offer a talk, using Maple, called Fermat's little theorem . It was never my intention to cover all of my prepared talk in Klagenfurt, and, in the event, I covered less than 0.1% of what I actually prepared. My Maple worksheet (129KB) may be downloaded here , and a (large) html version of it may be downloaded here large because Maple converts all outputs to gif files, and there are 447 of those in the worksheet). I dedicated my lecture to Mark Daly - a former colleague, and friend - as a token of my regard for him. Klaus Barner of Kassel university, Germany, disputes the date of Fermat's birth, and interested persons ought to read his papers: Pierre de Fermat (1601? - 1665) , European Mathematics Society Newsletter No. 42, December 2001 How old did Fermat become? 2001

11. Iatco Sergiu. Simplification Of Prove Of Fermat Theorem
Iatco Sergiu. Simplification of prove of fermat theorem. R First timeI read about Fermat s last theorem when I was 15 years old. Just
http://lib.ru/TXT/fermat.txt
Fine HTML Printed version txt(Word,ÃÃÃ) gZip Lib.ru html
##### Iatco Sergiu. Simplification of prove of Fermat theorem
R.Moldova District Rascani Village Recea itsergiu@yahoo.com Date: 4 May 1999 Dear Sir, I am an amateur mathematician. First time I read about Fermat's last theorem when I was 15 years old. Just like other people from the beginning I dreamt to prove one day it. Last year I found out that A.Wiles and R.Taylor proved it. I read this proof and I found it (just like other people) too complex. I analysed the Fermat's last theorem and I succeed to simplify it as follows: Let have Fermat's equation: a n +b n =c n , where n>2 (1) Because c=p1*...*pt, where pi - prime number, equation (1) becomes: a n +b n n *...*pt n (2) If exist such pi for which a1 n n = pi n (3) has solutions then these solutions are also solutions for (2) Let r= p1*...*pi-1*pi+1*...*pt Multiplying (3) with r n we have: (r*a1) n +(r*b1) n = pi n , let a=r*a1 b=r*b1 a n +b n n *...*pt

12. Modular Arithmetic, Fermat Theorem, Carmichael Numbers - Numericana
Modular arithmetic. The generalized theorem of Fermat and its converse versions,including Carmichael numbers and stochastic primality testing.
home index units counting ... physics
##### Modular Arithmetic
Mathematics is the Queen of sciences,
and arithmetic the Queen of mathematics.

Carl Friedrich Gauss (1777-1855)
##### Related articles on this site:

fermat theorem. Question Fermat s Last Theorem did Fermat have proof or did heguess? Submitted 16 February, 1998 by Jerry Marcantel of Glenmora, LA USA.
##### Fermat's Last Theorem: did Fermat have proof or did he guess?
Submitted 16 February, 1998 by Jerry Marcantel of Glenmora, LA USA. I am a amateur mathematician. On PBS I saw a show on the guy Andrew Wiles who proved Fermat's Last Theorem. Some of the mathematicians "hinted" that Fermat did not have any proof. That Fermat just stated his theorem with on proof. Is this a general consensus in the math community?
16 February 1998 This is definitely not my field of expertise, but I am pretty sure that it is the consensus of the experts that Fermat was probably mistaken. First, the problem has received an enormous amount of attention over the centuries from the very best mathematicians. The fact that none of them found a short proof (supposing Fermat's proof was not much longer than the margin would hold this is suggested by Fermat's description of it as "truly wonderful"), nor indeed any proof at all, makes one wonder. Second, it seems clear that if Fermat had a proof, it must have been quite different than the one we have today. For the prerequisites, the mathematical concepts used in the proof, on which Wiles was able to build, had not even been developed in Fermat's time. Third, there are some shorter partial proofs, and I think even some short, seductive, but mistaken ones, that have been discovered over the years, some by good mathematicians. Possibly Fermat's proof was one of these. Mathematicians do often guess, but of course a guess is never a proof! The inspired guess leads the way, motivates and guides the hard struggle to construct a rigorous proof, but no honest mathematician would ever knowingly say he had proved something that he had only guessed. But a strong hunch can lead you to believe a conjecture is true, and then it is not too uncommon to overlook subtle logical flaws in the proof constructed to establish the guess beyond all doubt. Wiles himself at first fell victim to such an error, which, fortunately, he was able to repair.

 14. The View | From The University Of Vermont fermat theorem Mathematician To Launch PresidentÂs Lecture Series.By Lynda Majarian. Andrew Wiles, the Princeton professor whohttp://www.uvm.edu/theview/article.php?id=738

15. The Mathematics Of Fermat's Last Theorem
Charles Daney's treatise on fermat's last theorem. HTML, DVI and PS.
http://www.mbay.net/~cgd/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 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. Enter 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. Just select the FLT Tex Files Link to begin downloading. Another request I receive frequently is for even more detailed information about Wiles' proof. The best reference, of course, is Wiles' own paper, which can be found in the Annals of Mathematics (3), May 1995. Suffice to say, it is very difficult reading. And you'll probably find it only in a good university library.

16. Fermat's Little Theorem
With notes on Carmichael numbers and the life of R.D. Carmichael.
http://www.geocities.com/Paris/Rue/1861/FermLit.html
 Fermat's Little Theorem Â  The famous "Last Theorem" for which Fermat is best know by students is not used nearly so often as the one which is remembered as his "little" theorem.Â  The little theorem is often used in number theory in the testing of large primes and simply states that:Â  if p is a prime which does not divide a, then a p-1 =1 (mod p) .Â  In more simple language this says that if p is a prime that is not a factor of a, then when a is multiplied together p-1 times, and the result divided by p, we get a remainder of one.Â  For example,Â  if we use a=7 and p=3, the rule says that 7 divided by 3 will have a remainder of one.Â  In fact 49/3 does have a remainder of one. The theorem was first stated by Fermat in a letter in 1640 without a proof. Euler gave the first published proof in 1736. Here is a link to a proof of the theorem Â Â  The theorem is a one direction theorem, what mathematicians call "necessary, but not sufficient".Â  What that means is that although it is true for all primes, it is not true JUST for primes, and will sometimes be true for other numbers as well.Â  For example 3

17. Fermat's Last Theorem
fermat's last theorem. Number theory index. History Topics Index. Pierre de fermat died in 1665. Today we think of fermat as a number theorist, in fact as perhaps the most famous number theorist who
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Fermat's_last_theorem.htm
##### Fermat's last theorem
Number Theory Index History Topics Index
Pierre de Fermat
died in 1665. Today we think of Fermat as a number theorist, in fact as perhaps the most famous number theorist who ever lived. It is therefore surprising to find that Fermat was in fact a lawyer and only an amateur mathematician. Also surprising is the fact that he published only one mathematical paper in his life, and that was an anonymous article written as an appendix to a colleague's book.
There is a statue of Fermat and his muse in his home town of Toulouse:
(Click it to see a larger version)
Because Fermat refused to publish his work, his friends feared that it would soon be forgotten unless something was done about it. His son, Samuel undertook the task of collecting Fermat 's letters and other mathematical papers, comments written in books, etc. with the object of publishing his father's mathematical ideas. In this way the famous 'Last theorem' came to be published. It was found by Samuel written as a marginal note in his father's copy of Diophantus 's Arithmetica Fermat's Last Theorem states that x n y n z n has no non-zero integer solutions for x y and z when n Fermat wrote I have discovered a truly remarkable proof which this margin is too small to contain.

18. Fermat's Last Theorem Poetry Challenge
fermat's Last theorem Poetry Challenge. While you're here, you can check out my home page, or visit the UIC math department. A special greeting to all new visitors from the PBS home page! Author
http://raphael.math.uic.edu/~jeremy/poetry.htm
##### Fermat's Last Theorem Poetry Challenge
While you're here, you can check out my home page, or visit the UIC math department. A special greeting to all new visitors from the PBS home page! Check out Fermat's Last Tango , a new musical opening off broadway on November 21, 2000. The proof of Fermat's last theorem by Andrew Wiles and Richard Taylor was presented to an audience of over 300 people during a tenday conference at Boston University in August, 1995. At that conference, I issued a poetry challenge asking for occasional verse to celebrate the proof. While the authors' anonymity was preserved at the meeting, all things are revealed in time. If you would like to contribute to this poetry competition, please send your masterpiece to Jeremy Teitelbaum . The editor's decisions regarding suitability for publication in this forum are arbitrary, personal, and final. With thanks to all of the participants, here are the entries (in no particular order). Author: John Fitzgerald Fermat's last theorem
Is a puzzling queer one:
Squares of a plane
Wholely squared, aren't arcane;

19. NOVA Online | The Proof | Solving Fermat: Andrew Wiles
I was just browsing through the section of math books and I found this one book,which was all about one particular problem fermat s Last theorem.
http://www.pbs.org/wgbh/nova/proof/wiles.html
##### Solving Fermat: Andrew Wiles
Andrew Wiles devoted much of his entire career to proving Fermat's Last Theorem, the world's most famous mathematical problem. In 1993, he made front-page headlines when he announced a proof of the problem, but this was not the end of the story; an error in his calculation jeopardized his life's work. Andrew Wiles spoke to NOVA and described how he came to terms with the mistake, and eventually went on to achieve his life's ambition.
NOVA:
Many great scientific discoveries are the result of obsession, but in your case that obsession has held you since you were a child.
ANDREW WILES: I grew up in Cambridge in England, and my love of mathematics dates from those early childhood days. I loved doing problems in school. I'd take them home and make up new ones of my own. But the best problem I ever found, I found in my local public library. I was just browsing through the section of math books and I found this one book, which was all about one particular problem Fermat's Last Theorem. This problem had been unsolved by mathematicians for 300 years. It looked so simple, and yet all the great mathematicians in history couldn't solve it. Here was a problem, that I, a ten year old, could understand and I knew from that moment that I would never let it go. I had to solve it.
NOVA: Who was Fermat and what was his Last Theorem?

20. Fermat's Last Theorem
fermat's Last theorem. On October 28, 1997, The Proof. will be broadcast on PBS as a program in the Nova series. ( See local listings for exact times.) in Cambridge, England that he had proven fermat's Last theorem. Email zapped around the globe as mathematicians
http://www.ams.org/new-in-math/fermat.html
 Fermat's Last Theorem On October 28, 1997, The Proof will be broadcast on PBS as a program in the Nova series. (See local listings for exact times.) This program was originally broadcast in Britain in January 1996 in the BBC Horizon series under the title Fermat's Last Theorem . Visit the BBC Horizon web site for information about the program, including a transcript. A review of this program is available on e-MATH. On June 23, 1993, Andrew Wiles announced to his colleagues at a mathematics conference in Cambridge, England that he had proven Fermat's Last Theorem. Email zapped around the globe as mathematicians and others celebrated the news. Newspapers all over the world trumpeted the achievement, and since then there have been many articles written about the proof. A number of web pages devoted to Fermat's Last Theorem have been started, among them the following: MacTutor History of Mathematics page on Fermat's Last Theorem Yahoo page on Fermat's Last Theorem State University of New York at Albany Department of Mathematics gopherFermat's Last Theorem Cambridge University Department of Pure Mathematics and Mathematical Statistics gopherFermat's Last Theorem The Proof of Fermat's Last Theorem by R. Taylor and A. Wiles

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