"페르마 소수"의 두 판 사이의 차이

수학노트
둘러보기로 가기 검색하러 가기
 
(같은 사용자의 중간 판 3개는 보이지 않습니다)
1번째 줄: 1번째 줄:
 
==개요==
 
==개요==
  
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*  페르마소수란 <math>F_n= 2^{2^n}+1</math> 형태의 소수
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*   페르마소수란 <math>F_n= 2^{2^n}+1</math> 형태의 소수
 
** 3,5,17,257, 65537 다섯 가지만 알려져 있음.
 
** 3,5,17,257, 65537 다섯 가지만 알려져 있음.
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*  페르마는  <math>F_n= 2^{2^n}+1</math> 가 모두 소수일 것이라 추측하였으나, 후에 [[오일러(1707-1783)|오일러]]는 반례를 발견:<math>F_5=641 \times 6700417</math>
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*  페르마는  <math>F_n= 2^{2^n}+1</math> 가 모두 소수일 것이라 추측하였으나, 후에 [[오일러(1707-1783)|오일러]]는 반례를 발견:<math>F_5=641 \times 6700417</math>
  
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==정다각형의 작도==
 
==정다각형의 작도==
  
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* 정n각형이 자와 컴파스로 작도가능 <math>\iff</math> <math>n=2^k p_1 p_2 \cdots p_r</math>  (k ,r은 0이상의 정수, <math>p_1, p_2, \cdots, p_r</math> 은 서로 다른 페르마소수)
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* 정n각형이 자와 컴파스로 작도가능 <math>\iff</math> <math>n=2^k p_1 p_2 \cdots p_r</math> (k ,r은 0이상의 정수, <math>p_1, p_2, \cdots, p_r</math> 은 서로 다른 페르마소수)
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* [[정다각형의 작도]]와 [[가우스와 정17각형의 작도]] 항목을 참조
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* [[정다각형의 작도]]와 [[가우스와 정17각형의 작도]] 항목을 참조
  
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==역사==
 
==역사==
  
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* [[수학사 연표]]  
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* [[수학사 연표]]
  
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==관련된 항목들==
 
==관련된 항목들==
32번째 줄: 32번째 줄:
  
 
[[분류:소수]]
 
[[분류:소수]]
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== 노트 ==
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===말뭉치===
 +
# In fact, it is known that numbers of this form are not prime for values of n from 5 through 30, placing doubt on the existence of any Fermat primes for values of n > 4.<ref name="ref_edbed968">[https://www.britannica.com/science/Fermat-prime Fermat prime | mathematics]</ref>
 +
# Fermat conjectured in 1650 that every Fermat number is prime and Eisenstein in 1844 proposed as a problem the proof that there are an infinite number of Fermat primes (Ribenboim 1996, p. 88).<ref name="ref_21863434">[https://mathworld.wolfram.com/FermatPrime.html Fermat Prime -- from Wolfram MathWorld]</ref>
 +
# Based on these results, one might conjecture (as did Fermat) that all Fermat numbers are prime.<ref name="ref_77005ae2">[https://artofproblemsolving.com/wiki/index.php/Fermat_prime Art of Problem Solving]</ref>
 +
# To find the Fermat number F n for an integer n , you first find m = 2 n , and then calculate 2 m + 1.<ref name="ref_7c2c701e">[https://www.techtarget.com/whatis/definition/Fermat-prime Definition from WhatIs.com]</ref>
 +
# Surprisingly, Fermat primes arise in deciding whether a regular n-gon (a convex polygon with n equal sides) can be constructed with a compass and a straightedge.<ref name="ref_a529a8a6">[https://sites.millersville.edu/bikenaga/number-theory/fermat-numbers/fermat-numbers.pdf Fermat numbers]</ref>
 +
# No fermat primes beyond (cid:8)4 have been found.<ref name="ref_a8ac433d">[http://www.math.ualberta.ca/~isaac/math324/s12/fermat_numbers.pdf Math 324 summer 2012]</ref>
 +
# There are in(cid:12)nitely many distinct Fermat numbers, each of which is divisible by an odd prime, and since any two Fermat numbers are relatively prime, these odd primes must all be distinct.<ref name="ref_a8ac433d" />
 +
# + 1 is a Fermat number; such primes are called Fermat primes.<ref name="ref_44f407e6">[https://en.wikipedia.org/wiki/Fermat_number Fermat number]</ref>
 +
# From the second equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbers share a common integer factor greater than 1.<ref name="ref_44f407e6" />
 +
# The sum of the reciprocals of all the Fermat numbers (sequence A051158 OEIS) is irrational.<ref name="ref_44f407e6" />
 +
# Indeed, the first five Fermat numbers F 0 , ..., F 4 are easily shown to be prime.<ref name="ref_44f407e6" />
 +
# The only known Fermat primes are the first five Fermat numbers: F 0 =3, F 1 =5, F 2 =17, F 3 =257, and F 4 =65537.<ref name="ref_fbf82f86">[https://primes.utm.edu/glossary/xpage/FermatNumber.html The Prime Glossary: Fermat number]</ref>
 +
# It only takes two trial divisions to find this factor because Euler showed that every divisor of a Fermat number F n with n greater than 2 has the form k.2n+1+1 (exponent improved to n+2 by Lucas).<ref name="ref_fbf82f86" />
 +
# Now we know that all of the Fermat numbers are composite for the other n less than 31.<ref name="ref_fbf82f86" />
 +
# In other words, every prime of the form 2k + 1 (other than 2 = 20 + 1 ) is a Fermat number, and such primes are called Fermat primes.<ref name="ref_e81de8fb">[http://www.scientificlib.com/en/Mathematics/LX/FermatNumber.html Fermat number]</ref>
 +
# From the last equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbers share a common integer factor greater than 1.<ref name="ref_e81de8fb" />
 +
# The sum of the reciprocals of all the Fermat numbers (sequence A051158 in OEIS) is irrational.<ref name="ref_e81de8fb" />
 +
# Indeed, the first five Fermat numbers F 0 ,...,F 4 are easily shown to be prime.<ref name="ref_e81de8fb" />
 +
# Indeed, even today, no other Fermat numbers are known to be prime!<ref name="ref_c596120d">[https://johncarlosbaez.wordpress.com/2019/02/05/fermat-primes-and-pascals-triangle/ Fermat Primes and Pascal’s Triangle]</ref>
 +
# Also show that every product of distinct Fermat numbers corresponds to a row of Pascal’s triangle mod 2.<ref name="ref_c596120d" />
 +
# Now, Gauss showed that we can construct a regular n-gon using straight-edge and compass if n is a prime Fermat number.<ref name="ref_c596120d" />
 +
# Wantzel went further and showed that if n is odd, we can construct a regular n-gon using straight-edge and compass if and only if n is a product of distinct Fermat primes.<ref name="ref_c596120d" />
 +
# There are two definitions of the Fermat number.<ref name="ref_25205313">[https://mathworld.wolfram.com/FermatNumber.html Fermat Number -- from Wolfram MathWorld]</ref>
 +
# The much more commonly encountered Fermat numbers are a special case, given by the binomial number of the form .<ref name="ref_25205313" />
 +
# Fermat conjectured in 1650 that every Fermat number is prime and Eisenstein proposed as a problem in 1844 the proof that there are an infinite number of Fermat primes (Ribenboim 1996, p. 88).<ref name="ref_25205313" />
 +
# At present, however, only composite Fermat numbers are known for .<ref name="ref_25205313" />
 +
# In other words, every prime of the form 2 n +1 is a Fermat number, and such primes are called Fermat primes.<ref name="ref_17d186c1">[https://www.rieselprime.de/ziki/Fermat_number Fermat number]</ref>
 +
# From the last equation, we can deduce Goldbach's theorem: no two Fermat numbers share a common factor.<ref name="ref_17d186c1" />
 +
# Indeed, the first five Fermat numbers F 0 ,..., F 4 are easily shown to be prime.<ref name="ref_17d186c1" />
 +
# Although it is widely believed that there are only finitely many Fermat primes, it should be noted that there are some experts who disagree (John Cosgrave: "Fermat 6").<ref name="ref_17d186c1" />
 +
# A019434 List of Fermat primes: primes of form, for someIt is conjectured that there are only 5 terms.<ref name="ref_b3f9ea5b">[https://oeis.org/wiki/Fermat_primes Fermat primes]</ref>
 +
# Numbers of the form F n =22n+1 are now called Fermat numbers*, and when they’re prime, they’re called Fermat primes.<ref name="ref_bfc52e99">[https://blogs.scientificamerican.com/roots-of-unity/extrapolation-gone-wrong-the-case-of-the-fermat-primes/ Extrapolation Gone Wrong: the Case of the Fermat Primes]</ref>
 +
# Fermat conjectured that all Fermat numbers are prime.<ref name="ref_bfc52e99" />
 +
# In 1732, about 70 years after Fermat's death, Leonhard Euler factored the 5th Fermat number into 641×6,700,417, disproving Fermat’s conjecture.<ref name="ref_bfc52e99" />
 +
# So far, the only known Fermat primes are the ones that were known to Fermat.<ref name="ref_bfc52e99" />
 +
# Is there a formula or a way to know which of the Fermat numbers are prime?.<ref name="ref_dc7c427d">[https://math.stackexchange.com/questions/563390/fermat-numbers-are-they-all-prime Fermat numbers. Are they all prime?]</ref>
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# Prologue What are the known Fermat primes?<ref name="ref_b2c9d49a">[https://arxiv.org/pdf/1605.01371 This is the extended version of a paper that has appeared in the Mathematical Intelligencer. The final publication is available]</ref>
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# Taking Fermat prime to mean prime of the form 2n + 1, there are six known Fermat primes, namely those for n = 0, 1, 2, 4, 8, 16.<ref name="ref_b2c9d49a" />
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# We shall pronounce the last letter of Fermats name, as he did, when we include 2 among the Fermat primes, as he did.<ref name="ref_b2c9d49a" />
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# The Fermat number Fn is either prime or not prime: the question of how to approximate the probability of primality for a general n is delicate.<ref name="ref_b2c9d49a" />
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# Fb;n = b2n + 1 and are particularly interesting since they have many characteristics of the heavily studied standard Fermat numbers Fn = F2;n.<ref name="ref_a10cb56f">[https://www.ams.org/journals/mcom/2002-71-238/S0025-5718-01-01350-3/S0025-5718-01-01350-3.pdf Mathematics of computation]</ref>
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# The \Proth" program was created in 1997 to extend the search for large factors of Fermat numbers.<ref name="ref_a10cb56f" />
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# They are called Fermat numbers, named after the French mathematician Pierre de Fermat (1601 1665) who first studied numbers in this form.<ref name="ref_8f117ada">[https://wstein.org/edu/2010/414/projects/tsang.pdf Fermat]</ref>
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# We will not be able to answer this question in this paper, but we will prove some basic properties of Fermat numbers and discuss their primality and divisibility.<ref name="ref_8f117ada" />
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# Primes in this form are called Fermat primes.<ref name="ref_8f117ada" />
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# Up-to-date there are only five known Fermat primes.<ref name="ref_8f117ada" />
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# Moreover, no other Fermat number is known to be prime for n > 4 , so now it is conjectured that those are all prime Fermat numbers.<ref name="ref_6428f90c">[https://planetmath.org/fermatnumbers Fermat numbers]</ref>
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# In honour of the inspired pioneers, the numbers of the form 2n-1 are now called the Mersenne numbers and the numbers of the form 2n+1 the Fermat numbers.<ref name="ref_e35d89c0">[http://yves.gallot.pagesperso-orange.fr/primes/ Generalized Fermat Primes Search]</ref>
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# The search for Mersenne and Fermat primes has been greatly extended since the 17th century.<ref name="ref_e35d89c0" />
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# Today, all the Mersenne primes having less than 2,000,000 digits are known and all the Fermat primes up to 2,000,000,000 digits!<ref name="ref_e35d89c0" />
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# In 1994, R. Crandall and B. Fagin discovered that the Discrete Weighted Transforms could be used to double the speed of the search for Mersenne and Fermat numbers.<ref name="ref_e35d89c0" />
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# Those are the only primes below 100,000 that I could show must be primitive for all but finitely many Fermat primes.<ref name="ref_9d437e74">[https://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_putnam;action=display;num=1204796915 Fermat Primes and the number 7]</ref>
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# Fermat numbers are named after Pierre de Fermat.<ref name="ref_a462b554">[https://kids.kiddle.co/Fermat_number Fermat number facts for kids]</ref>
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# Every prime of the form 2n + 1 is a Fermat number, and such primes are called Fermat primes.<ref name="ref_a462b554" />
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# Fermat numbers can be calculated recursively: To get the Nth number, multiply all Fermat numbers before it, and add two to the result.<ref name="ref_a462b554" />
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# Two previous posts (here and here) present an alternative proof that there are infinitely many prime numbers using the Fermat numbers.<ref name="ref_e5ed5592">[https://exploringnumbertheory.wordpress.com/2016/10/25/fermat-numbers/ Exploring Number Theory]</ref>
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# Specifically, the proof is accomplished by pointing out that the prime factors of the Fermat numbers form an infinite set.<ref name="ref_e5ed5592" />
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# The numbers grow very rapidly since each Fermat number is obtained by raising 2 to a power of 2.<ref name="ref_e5ed5592" />
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# He demonstrated that the first 5 Fermat numbers , , , , are prime and conjectured that all Fermat numbers are prime.<ref name="ref_e5ed5592" />
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# Chapter 4 Fermat and Mersenne Primes 4.1 Fermat primes Theorem 4.1.<ref name="ref_7befcda2">[https://www.maths.tcd.ie/pub/Maths/Courseware/NumberTheory/FermatMersenne.pdf Chapter 4]</ref>
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# Fermat conjectured that the Fermat numbers are all prime.<ref name="ref_7befcda2" />
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# We will come back to their solution shortly as we must first introduce the notion of a Fermat prime!<ref name="ref_62f9f1ec">[https://vrs.amsi.org.au/fermat-primes-gauss-wantzel/ Fermat Primes and the Gauss-Wantzel Theorem]</ref>
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# In the 1600s, a mathematician and lawyer named Pierre de Fermat studied numbers of the form 2^n+1 (where n=2^k) which are now called Fermat numbers.<ref name="ref_62f9f1ec" />
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# So, he conjectured that all Fermat numbers are prime.<ref name="ref_62f9f1ec" />
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# Funnily enough, it was shown by Leonhard Euler in 1732 (another famous mathematician), that only the next Fermat number is not prime.<ref name="ref_62f9f1ec" />
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# It is still open whether there exist innitely many Fermat primes or innitely many composite Fermat numbers.<ref name="ref_0b05fdb4">[https://arxiv.org/pdf/2204.08302 MINIMALITY CONDITIONS EQUIVALENT TO THE FINITUDE OF FERMAT AND MERSENNE PRIMES]</ref>
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# Especially, the formula for modulo Fermat primes is given. MSC2010: 11A07.<ref name="ref_9c963bc5">[https://arxiv.org/pdf/1601.06509 The largest cycles consist by the quadratic residues and Fermat primes]</ref>
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# For the Fermat primes, i.e., the prime numbers of the form p 22k ` 1, we have Lppq 1. Proof.<ref name="ref_9c963bc5" />
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# We have veried it for k 0, 1, 2, 3, 4, the known Fermat primes.<ref name="ref_9c963bc5" />
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# As of 2021, the only known Fermat primes are F0 = 3, F1 = 5, F2 = 17, F3 = 257, and F4 = 65537.<ref name="ref_49cdcbc7">[https://arxiv.org/pdf/1912.12088 MINIMALITY OF TOPOLOGICAL MATRIX GROUPS AND FERMAT PRIMES]</ref>
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# For an odd prime p the following conditions are equivalent: (1) p is a Fermat prime; (2) SL(p 1, (Q, p)) is minimal; (3) SL(p 1, Q(i)) is minimal.<ref name="ref_49cdcbc7" />
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# We determine all the Carmichael numbers m with a Fermat prime factor such that L = 2P 2, where k N and P is an odd prime number.<ref name="ref_f1f90697">[https://arxiv.org/pdf/1710.01321 On the finiteness of Carmichael numbers with Fermat factors and L = 2αP 2.]</ref>
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# We assume that m is divisible by at least one of the known Fermat prime numbers.<ref name="ref_f1f90697" />
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# If m is divisible by one of the known Fermat primes, then m must be one of the following 11 Carmichael numbers.<ref name="ref_f1f90697" />
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# Let 2 R R 22, there exists a generalized Fermat prime p = r2 2 be an integer.<ref name="ref_25a71c73">[https://arxiv.org/pdf/1502.02800 FAST INTEGER MULTIPLICATION USING GENERALIZED FERMAT PRIMES]</ref>
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# The key concept of our algorithm is the use of a chain of generalized Fermat primes (of the form r2 +1) to handle recursive calls.<ref name="ref_25a71c73" />
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# First, we encode integers to be multiplied as integers modulo generalized Fermat primes, and not as polynomials.<ref name="ref_25a71c73" />
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# Section 4 studies generalized Fermat primes, and their relation to the Bateman-Horn conjecture.<ref name="ref_25a71c73" />
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# 1. Introduction + 1 for n The Fermat numbers are given by Fn = 22n 0.<ref name="ref_4a2a6fd4">[https://arxiv.org/pdf/2102.00906 ON UPPER BOUNDS FOR THE COUNT OF ELITE PRIMES Matthew Just]</ref>
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# Notice that the rst ve Fermat numbers are prime, and it was initially conjectured (by Fermat) that all such num- bers are prime.<ref name="ref_4a2a6fd4" />
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# The sixth Fermat number is not prime, and no other Fermat primes are known.<ref name="ref_4a2a6fd4" />
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# An ecient test exists to determine whether or not a Fermat number is prime, called Pepins test.<ref name="ref_4a2a6fd4" />
 +
# This characterization uses uniquely values at most equal to tested Fermat number.<ref name="ref_bca905b7">[https://arxiv.org/pdf/2104.04875 A NEW CHARACTERIZATION OF PRIME FERMAT’S NUMBERS]</ref>
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# We actually are able to establish the direct implication which is a real important result in the primality tests for Fermat numbers.<ref name="ref_bca905b7" />
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# Condition 2m + = pn requires p to be either a Mersenne or Fermat prime.<ref name="ref_4393ff84">[https://arxiv.org/pdf/1809.03328 On Upper Bounds with ABC = 2mpn and ABC = 2mpnqr with p and q as Mersenne or Fermat]</ref>
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===소스===
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<references />
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== 메타데이터 ==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q207264 Q207264]
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===Spacy 패턴 목록===
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* [{'LOWER': 'fermat'}, {'LEMMA': 'prime'}]
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* [{'LOWER': 'fermat'}, {'LEMMA': 'number'}]

2022년 9월 16일 (금) 02:47 기준 최신판

개요

  • 페르마소수란 \(F_n= 2^{2^n}+1\) 형태의 소수
    • 3,5,17,257, 65537 다섯 가지만 알려져 있음.
  • 페르마는 \(F_n= 2^{2^n}+1\) 가 모두 소수일 것이라 추측하였으나, 후에 오일러는 반례를 발견\[F_5=641 \times 6700417\]



정다각형의 작도




역사


관련된 항목들

노트

말뭉치

  1. In fact, it is known that numbers of this form are not prime for values of n from 5 through 30, placing doubt on the existence of any Fermat primes for values of n > 4.[1]
  2. Fermat conjectured in 1650 that every Fermat number is prime and Eisenstein in 1844 proposed as a problem the proof that there are an infinite number of Fermat primes (Ribenboim 1996, p. 88).[2]
  3. Based on these results, one might conjecture (as did Fermat) that all Fermat numbers are prime.[3]
  4. To find the Fermat number F n for an integer n , you first find m = 2 n , and then calculate 2 m + 1.[4]
  5. Surprisingly, Fermat primes arise in deciding whether a regular n-gon (a convex polygon with n equal sides) can be constructed with a compass and a straightedge.[5]
  6. No fermat primes beyond (cid:8)4 have been found.[6]
  7. There are in(cid:12)nitely many distinct Fermat numbers, each of which is divisible by an odd prime, and since any two Fermat numbers are relatively prime, these odd primes must all be distinct.[6]
  8. + 1 is a Fermat number; such primes are called Fermat primes.[7]
  9. From the second equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbers share a common integer factor greater than 1.[7]
  10. The sum of the reciprocals of all the Fermat numbers (sequence A051158 OEIS) is irrational.[7]
  11. Indeed, the first five Fermat numbers F 0 , ..., F 4 are easily shown to be prime.[7]
  12. The only known Fermat primes are the first five Fermat numbers: F 0 =3, F 1 =5, F 2 =17, F 3 =257, and F 4 =65537.[8]
  13. It only takes two trial divisions to find this factor because Euler showed that every divisor of a Fermat number F n with n greater than 2 has the form k.2n+1+1 (exponent improved to n+2 by Lucas).[8]
  14. Now we know that all of the Fermat numbers are composite for the other n less than 31.[8]
  15. In other words, every prime of the form 2k + 1 (other than 2 = 20 + 1 ) is a Fermat number, and such primes are called Fermat primes.[9]
  16. From the last equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbers share a common integer factor greater than 1.[9]
  17. The sum of the reciprocals of all the Fermat numbers (sequence A051158 in OEIS) is irrational.[9]
  18. Indeed, the first five Fermat numbers F 0 ,...,F 4 are easily shown to be prime.[9]
  19. Indeed, even today, no other Fermat numbers are known to be prime![10]
  20. Also show that every product of distinct Fermat numbers corresponds to a row of Pascal’s triangle mod 2.[10]
  21. Now, Gauss showed that we can construct a regular n-gon using straight-edge and compass if n is a prime Fermat number.[10]
  22. Wantzel went further and showed that if n is odd, we can construct a regular n-gon using straight-edge and compass if and only if n is a product of distinct Fermat primes.[10]
  23. There are two definitions of the Fermat number.[11]
  24. The much more commonly encountered Fermat numbers are a special case, given by the binomial number of the form .[11]
  25. Fermat conjectured in 1650 that every Fermat number is prime and Eisenstein proposed as a problem in 1844 the proof that there are an infinite number of Fermat primes (Ribenboim 1996, p. 88).[11]
  26. At present, however, only composite Fermat numbers are known for .[11]
  27. In other words, every prime of the form 2 n +1 is a Fermat number, and such primes are called Fermat primes.[12]
  28. From the last equation, we can deduce Goldbach's theorem: no two Fermat numbers share a common factor.[12]
  29. Indeed, the first five Fermat numbers F 0 ,..., F 4 are easily shown to be prime.[12]
  30. Although it is widely believed that there are only finitely many Fermat primes, it should be noted that there are some experts who disagree (John Cosgrave: "Fermat 6").[12]
  31. A019434 List of Fermat primes: primes of form, for someIt is conjectured that there are only 5 terms.[13]
  32. Numbers of the form F n =22n+1 are now called Fermat numbers*, and when they’re prime, they’re called Fermat primes.[14]
  33. Fermat conjectured that all Fermat numbers are prime.[14]
  34. In 1732, about 70 years after Fermat's death, Leonhard Euler factored the 5th Fermat number into 641×6,700,417, disproving Fermat’s conjecture.[14]
  35. So far, the only known Fermat primes are the ones that were known to Fermat.[14]
  36. Is there a formula or a way to know which of the Fermat numbers are prime?.[15]
  37. Prologue What are the known Fermat primes?[16]
  38. Taking Fermat prime to mean prime of the form 2n + 1, there are six known Fermat primes, namely those for n = 0, 1, 2, 4, 8, 16.[16]
  39. We shall pronounce the last letter of Fermats name, as he did, when we include 2 among the Fermat primes, as he did.[16]
  40. The Fermat number Fn is either prime or not prime: the question of how to approximate the probability of primality for a general n is delicate.[16]
  41. Fb;n = b2n + 1 and are particularly interesting since they have many characteristics of the heavily studied standard Fermat numbers Fn = F2;n.[17]
  42. The \Proth" program was created in 1997 to extend the search for large factors of Fermat numbers.[17]
  43. They are called Fermat numbers, named after the French mathematician Pierre de Fermat (1601 1665) who first studied numbers in this form.[18]
  44. We will not be able to answer this question in this paper, but we will prove some basic properties of Fermat numbers and discuss their primality and divisibility.[18]
  45. Primes in this form are called Fermat primes.[18]
  46. Up-to-date there are only five known Fermat primes.[18]
  47. Moreover, no other Fermat number is known to be prime for n > 4 , so now it is conjectured that those are all prime Fermat numbers.[19]
  48. In honour of the inspired pioneers, the numbers of the form 2n-1 are now called the Mersenne numbers and the numbers of the form 2n+1 the Fermat numbers.[20]
  49. The search for Mersenne and Fermat primes has been greatly extended since the 17th century.[20]
  50. Today, all the Mersenne primes having less than 2,000,000 digits are known and all the Fermat primes up to 2,000,000,000 digits![20]
  51. In 1994, R. Crandall and B. Fagin discovered that the Discrete Weighted Transforms could be used to double the speed of the search for Mersenne and Fermat numbers.[20]
  52. Those are the only primes below 100,000 that I could show must be primitive for all but finitely many Fermat primes.[21]
  53. Fermat numbers are named after Pierre de Fermat.[22]
  54. Every prime of the form 2n + 1 is a Fermat number, and such primes are called Fermat primes.[22]
  55. Fermat numbers can be calculated recursively: To get the Nth number, multiply all Fermat numbers before it, and add two to the result.[22]
  56. Two previous posts (here and here) present an alternative proof that there are infinitely many prime numbers using the Fermat numbers.[23]
  57. Specifically, the proof is accomplished by pointing out that the prime factors of the Fermat numbers form an infinite set.[23]
  58. The numbers grow very rapidly since each Fermat number is obtained by raising 2 to a power of 2.[23]
  59. He demonstrated that the first 5 Fermat numbers , , , , are prime and conjectured that all Fermat numbers are prime.[23]
  60. Chapter 4 Fermat and Mersenne Primes 4.1 Fermat primes Theorem 4.1.[24]
  61. Fermat conjectured that the Fermat numbers are all prime.[24]
  62. We will come back to their solution shortly as we must first introduce the notion of a Fermat prime![25]
  63. In the 1600s, a mathematician and lawyer named Pierre de Fermat studied numbers of the form 2^n+1 (where n=2^k) which are now called Fermat numbers.[25]
  64. So, he conjectured that all Fermat numbers are prime.[25]
  65. Funnily enough, it was shown by Leonhard Euler in 1732 (another famous mathematician), that only the next Fermat number is not prime.[25]
  66. It is still open whether there exist innitely many Fermat primes or innitely many composite Fermat numbers.[26]
  67. Especially, the formula for modulo Fermat primes is given. MSC2010: 11A07.[27]
  68. For the Fermat primes, i.e., the prime numbers of the form p 22k ` 1, we have Lppq 1. Proof.[27]
  69. We have veried it for k 0, 1, 2, 3, 4, the known Fermat primes.[27]
  70. As of 2021, the only known Fermat primes are F0 = 3, F1 = 5, F2 = 17, F3 = 257, and F4 = 65537.[28]
  71. For an odd prime p the following conditions are equivalent: (1) p is a Fermat prime; (2) SL(p 1, (Q, p)) is minimal; (3) SL(p 1, Q(i)) is minimal.[28]
  72. We determine all the Carmichael numbers m with a Fermat prime factor such that L = 2P 2, where k N and P is an odd prime number.[29]
  73. We assume that m is divisible by at least one of the known Fermat prime numbers.[29]
  74. If m is divisible by one of the known Fermat primes, then m must be one of the following 11 Carmichael numbers.[29]
  75. Let 2 R R 22, there exists a generalized Fermat prime p = r2 2 be an integer.[30]
  76. The key concept of our algorithm is the use of a chain of generalized Fermat primes (of the form r2 +1) to handle recursive calls.[30]
  77. First, we encode integers to be multiplied as integers modulo generalized Fermat primes, and not as polynomials.[30]
  78. Section 4 studies generalized Fermat primes, and their relation to the Bateman-Horn conjecture.[30]
  79. 1. Introduction + 1 for n The Fermat numbers are given by Fn = 22n 0.[31]
  80. Notice that the rst ve Fermat numbers are prime, and it was initially conjectured (by Fermat) that all such num- bers are prime.[31]
  81. The sixth Fermat number is not prime, and no other Fermat primes are known.[31]
  82. An ecient test exists to determine whether or not a Fermat number is prime, called Pepins test.[31]
  83. This characterization uses uniquely values at most equal to tested Fermat number.[32]
  84. We actually are able to establish the direct implication which is a real important result in the primality tests for Fermat numbers.[32]
  85. Condition 2m + = pn requires p to be either a Mersenne or Fermat prime.[33]

소스

  1. ↑ Fermat prime | mathematics
  2. ↑ Fermat Prime -- from Wolfram MathWorld
  3. ↑ Art of Problem Solving
  4. ↑ Definition from WhatIs.com
  5. ↑ Fermat numbers
  6. ↑ 6.0 6.1 Math 324 summer 2012
  7. ↑ 7.0 7.1 7.2 7.3 Fermat number
  8. ↑ 8.0 8.1 8.2 The Prime Glossary: Fermat number
  9. ↑ 9.0 9.1 9.2 9.3 Fermat number
  10. ↑ 10.0 10.1 10.2 10.3 Fermat Primes and Pascal’s Triangle
  11. ↑ 11.0 11.1 11.2 11.3 Fermat Number -- from Wolfram MathWorld
  12. ↑ 12.0 12.1 12.2 12.3 Fermat number
  13. ↑ Fermat primes
  14. ↑ 14.0 14.1 14.2 14.3 Extrapolation Gone Wrong: the Case of the Fermat Primes
  15. ↑ Fermat numbers. Are they all prime?
  16. ↑ 16.0 16.1 16.2 16.3 This is the extended version of a paper that has appeared in the Mathematical Intelligencer. The final publication is available
  17. ↑ 17.0 17.1 Mathematics of computation
  18. ↑ 18.0 18.1 18.2 18.3 Fermat
  19. ↑ Fermat numbers
  20. ↑ 20.0 20.1 20.2 20.3 Generalized Fermat Primes Search
  21. ↑ Fermat Primes and the number 7
  22. ↑ 22.0 22.1 22.2 Fermat number facts for kids
  23. ↑ 23.0 23.1 23.2 23.3 Exploring Number Theory
  24. ↑ 24.0 24.1 Chapter 4
  25. ↑ 25.0 25.1 25.2 25.3 Fermat Primes and the Gauss-Wantzel Theorem
  26. ↑ MINIMALITY CONDITIONS EQUIVALENT TO THE FINITUDE OF FERMAT AND MERSENNE PRIMES
  27. ↑ 27.0 27.1 27.2 The largest cycles consist by the quadratic residues and Fermat primes
  28. ↑ 28.0 28.1 MINIMALITY OF TOPOLOGICAL MATRIX GROUPS AND FERMAT PRIMES
  29. ↑ 29.0 29.1 29.2 On the finiteness of Carmichael numbers with Fermat factors and L = 2αP 2.
  30. ↑ 30.0 30.1 30.2 30.3 FAST INTEGER MULTIPLICATION USING GENERALIZED FERMAT PRIMES
  31. ↑ 31.0 31.1 31.2 31.3 ON UPPER BOUNDS FOR THE COUNT OF ELITE PRIMES Matthew Just
  32. ↑ 32.0 32.1 A NEW CHARACTERIZATION OF PRIME FERMAT’S NUMBERS
  33. ↑ On Upper Bounds with ABC = 2mpn and ABC = 2mpnqr with p and q as Mersenne or Fermat


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