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Topic: Fermat number

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In the News (Tue 23 Jul 19)

  Pierre de Fermat - Wikipedia, the free encyclopedia
He created the two-square theorem, and the polygonal number theorem, which states that each number is a sum of 3 triangular numbers, 4 square numbers, 5 pentagonal numbers,...
Together with René Descartes, Fermat was one of the two leading mathematicians of the first half of the 17th century.
Fermat was born at Beaumont-de-Lomagne, 58 kilometers (36 miles) north-west of Toulouse, France.
en.wikipedia.org /wiki/Fermat   (648 words)

 PlanetMath: Fermat Numbers
Fermat incorrectly conjectured that all these numbers were primes, although he had no proof.
It is also unknown whether there are infinitely many composite Fermat numbers or not.
This is version 13 of Fermat Numbers, born on 2001-08-26, modified 2006-10-26.
planetmath.org /encyclopedia/FermatNumbers.html   (178 words)

 Pierre de Fermat Summary
Fermat refused to publish his work except in the form of challenges to other mathematicians of problems and theorems to be solved.
Fermat developed numerous theorems involving prime numbers and integral numbers and, consequently, is regarded as the founder of modern number theory.
Fermat, who added the aristocratic "de" to his name in his early 1630s, was the son of Dominique Fermat, a successful leather merchant, and Claire de Long, who came from a highly respected family of lawyers.
www.bookrags.com /Pierre_de_Fermat   (5076 words)

 Fermat number Summary
Fermat numbers and Fermat primes were first studied by Pierre de Fermat, who conjectured that all Fermat numbers are prime.
One common explanation is that Fermat made a computational mistake and was so convinced of the correctness of his claim that he failed to double-check his work.
Because of the size of Fermat numbers, it is difficult to factorize or to prove primality of those.
www.bookrags.com /Fermat_number   (1320 words)

 Ivars Peterson's MathTrek - Cracking Fermat Numbers
Fermat numbers have what mathematicians sometimes describe as a "beautiful mathematical form," involving powers of 2.
As of Feb. 25, 212 Fermat numbers were known to be composite, and searchers had found a total of 245 prime factors.
The divisor itself is the fifth largest known prime number, and it is the largest that is not a Mersenne prime.
www.maa.org /mathland/mathtrek_03_03_03.html   (591 words)

 Number Theory - Euler
Number Theory is the area of mathematics concerned primarily with integer (sometimes rational) solutions to expressions.
Fermat later rediscovered this pair, to be shortly thereafter outdone by Rene Descartes with (9 363 584, 9 437 056).
Fermat "prime" is not at all prime, and this one counterexample shatters Fermat's far-reaching conjecture.
members.aol.com /tylern7/math/euler-6.html   (1564 words)

 All Fermat numbers are squarefree (and more)
Since Fermat numbers are the "least random numbers of all numbers" (a sequence of 0s limited by two 1s), the weight of such arguments has to be considered prudentially.
Antisymmetric numbers are numbers whose expression 1/n in binary is a ß/2 long sequence followed by the same sequence inverted.
Sentence 7 is "one of the many" properties of factors of Fermat numbers that suggests an explanation of why they are hard to factor even if their ß is known.
www.dybot.com /numbers/sqfree.htm   (4373 words)

 Fermat number - Wikipedia, the free encyclopedia
The sum of the reciprocals of the Fermat numbers is irrational.
Fermat primes are particularly useful in generating pseudorandom sequences of numbers that fit the form 1 - N where N is a power of 2.
Now multiply this by a relatively prime number greater than the square root of P, and take the result MOD P. The result is the new value for the RNG.
en.wikipedia.org /wiki/Fermat_number   (1410 words)

 The Prime Glossary: Fermat number
So we call these the Fermat numbers, and when a number of this form is prime, we call it a Fermat prime.
Now we know that all of the Fermat numbers are composite for the other n less than 31.
The quickest way to check a Fermat number for primality (if trial division fails to find a small factor) is to use Pepin's test.
primes.utm.edu /glossary/page.php?sort=FermatNumber   (551 words)

 Fermat's last theorem - an elementary proof by Nico de Jong (1992)
An equation in three natural number terms is presented as the greatest term being equated to the sum of the two other terms.
The Fibonacci identity proves that a number of primitive Pythagorean equations are constructible from the squares of the prime factors of z, if all the prime factors of z are of the q-type.
This means that any effort to assign natural number values to a and b, and therefore to the last prime in z descended to, will change its value and with it the value of the original z.
www.geocities.com /elementaryfermat   (5445 words)

 [No title]
Fermat surely knew that when n = 1 the two numbers (34 and 20) are not amicable but he failed to mention such in his letter to Mersenne, perhaps dismissing the counterexample as obvious.
Fermat then announced the solution to the problem: “Clearly, the number of different pairs of squares of which N represents the difference depends on the number of different pairs of odd factors of which N is composed.” [7, p.
Fermat’s factoring algorithm was the first to improve upon the Sieve of Eratosthanes and it remained the algorithm of choice for hundreds of years to follow.
www.math.utexas.edu /~narula/fermat.doc   (5434 words)

 Sequence A000215, Fermat Numbers at MROB
The Fermat numbers, Sloane's A000215, are numbers of the form 2
In the years since the Fermat numbers and their factors have been the subject of much research.
This limits the number of primes that must be tested to factor a given Fermat number.
home.earthlink.net /~mrob/pub/math/seq-a000215.html   (208 words)

 Fermat 6
Here, then, is my discovered linking of the Mersenne and Fermat numbers (if it is already buried out there in the literature, then obviously I apologise for wasting your time in reading this, and I end up with egg on face.
The Mersenne numbers are the M[p] with M[p] = 2^p - 1, p prime; the Fermat numbers are the F[n] = 2^(2^n) + 1, n = 0, 1, 2, 3, 4, 5,...
But the Fermat numbers are simply the left vertical side of the above square, and what about the next rank along from it?, and the one alongside it?, and...
services.spd.dcu.ie /johnbcos/fermat6.htm   (1986 words)

 The Prime Glossary: generalized Fermat number
(with n and b integers, b greater than one) are called the generalized Fermat numbers because they are Fermat numbers in the special case b=2.
When b is even, these numbers share many properties with the regular Fermat numbers.
On the rare occasion that these generalized Fermat numbers are prime, they are call generalized Fermat primes.
primes.utm.edu /glossary/page.php?sort=GeneralizedFermatNumber   (140 words)

 A Short-Form Proof of Fermat's Last Theorem
Validating Fermats assertion that he had a proof is of utmost importance.
We have no direct knowledge of what might have been Fermats general approach; however, we have a few tantalizing clues from the notes he is said to have made in the margin of a text.
The root number, 2(3*1), which is 6, yields two valid solutions since the prime, 3, may be transposed from the even to the odd factor.
www.fermatproof.com   (3281 words)

 Generalized Fermat Primes Search   (Site not responding. Last check: 2007-11-03)
Fermat knew that 3, 5, 17, 257 and 65537 are primes but later Euler showed that Fermat's conjecture is false by discovering a factor to the next number.
But the Fermat and Mersenne primes are rare and the chance to find a new prime is small.
Generalized Fermat Numbers are more numerous than Mersenne numbers at equal size and many of them are waiting to be discovered to fill the gaps between the Mersenne primes already found or not yet found.
perso.wanadoo.fr /yves.gallot/primes   (679 words)

 Math Trek: Cracking Fermat Numbers, Science News Online, March 1, 2003   (Site not responding. Last check: 2007-11-03)
What's striking about the sequence is the rapidity with which the size of the numbers grows larger.
With the advent of computers and, recently, a concerted effort to use the spare processing power of computers around the world to test for divisors of Fermat numbers (see http://www.fermatsearch.org/), the search for factors has expanded considerably.
John B. Cosgrove describes the discovery of the largest known composite Fermat number at http://www.spd.dcu.ie/johnbcos/.
www.sciencenews.org /20030301/mathtrek.asp   (703 words)

 Pierre de Fermat — Infoplease.com
A magistrate whose avocation was mathematics, Fermat is known as a founder of modern
) and invented a number of methods for determining maxima and minima that were later of use to Newton in applying the calculus.
number theory - number theory, branch of mathematics concerned with the properties of the integers (the numbers 0,...
www.infoplease.com /ce6/people/A0818496.html   (347 words)

 Large Numbers -- Notes at MROB
Their goal was to make it possible to derive every true theorem in number theory by starting with a set of axioms and a set of inference rules, and methodically applying all the inference rules to the axioms and existing theorems to create new theorems.
But the number of 1's increases in a way that also depends in its value modulo 4, with the result that the value modulo 4 changes in a "chaotic" manner.
If the number of terms is infinite, P has to be defined a different way, because the infinite regression symbolized by "..." is not allowed at the beginning of an infinite ordinal sum (for technical reasons).
home.earthlink.net /~mrob/pub/math/ln-notes1.html   (8488 words)

 Fermat Number Record
Complete details concerning the current state of knowledge concerning Fermat numbers (put together by Wilfrid Keller, himself a discoverer of many Fermat number results) are available at this location.
I went downstairs to tell my wife that we had almost certainly found the largest composite Fermat number, otherwise Yves Gallot's letter simply would not be sensible...
However in Notepad the entire log of the computations was saved, and today my colleague Paul Murphy took a number of digital photos of the screen of computer #17 (appropriately '17' is a Fermat number!!).
services.spd.dcu.ie /johnbcos/fermat.htm   (1487 words)

 Conjecture 4. Fermat primes are finite.
It’s known that Pierre Fermat believed and tried to prove that all the Fermat numbers are primes, but he didn’t succeed.
After that we know that some Fermat numbers are primes (five) and others are composites.
He hopes then that then probably that statement is also not valid in the rank where these new class of numbers coincides with the classical Fermat numbers.
www.primepuzzles.net /conjectures/conj_004.htm   (1104 words)

 FAQ for Fermat numbers
) numbers are known to be prime or composite.
Though there are few practical applications of find Fermat divisors, but the algorithms used for finding them have applications to other computing tasks.
Finally, the most pleasing file is Results.txt that is only created when a Fermat number divider is found.
www.fermatsearch.org /faq.htm   (923 words)

The idelic approach to number theory, introduction to local fields, the modular curves X
Analytic Number Theory and Applications: Collection of papers on the occasion of the 60th birthday of Anatolli Alexeevich Karatsuba, Proc.
A Bibliography of Bernoulli Numbers (Karl Dilcher and Ilja Slavutskii)
www.numbertheory.org /ntw/N4.html   (2139 words)

 The On-Line Encyclopedia of Integer Sequences   (Site not responding. Last check: 2007-11-03)
For n>0, Fermat numbers F(n) have digital roots 5 or 8 depending on whether n is even or odd (Koshy).
G. Hardy and E. Wright, An Introduction to the Theory of Numbers.
Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m
www.research.att.com /projects/OEIS?Anum=A000215   (340 words)

 F. Conjectures (Math 413, Number Theory)
A collection of easily stated number theory conjectures which are still open.
Conj: Only a finite number of Fermat numbers are prime.
A stronger conjecture is that the only prime Fermat numbers are F
www.math.umbc.edu /~campbell/Math413Fall98/Conjectures.html   (485 words)

 Mathematics Archives - Topics in Mathematics - Number Theory
Elementary Number Theory, Lucas' Theorem, Pascal's triangle via cellular automata, Bernoulli numbers and polynomials, Theorems of Morley and Emma Lehmer and their generalizations, Some useful p-adic numbers
Class notes, congruences, Chinese remainder theorem, Fermat's Theorem, Primality testing, Miller-Rabin test, Cryptography, Mersenne numbers, quadratic residues, Diophantine equations, Pell's equation, continued fractions.
Divisibility and primes, Euclidean algorithm, Euler's theorem, Representation of numbers, Bertrand's postulate
archives.math.utk.edu /topics/numberTheory.html   (675 words)

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