페르마 소수

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

• 정n각형이 자와 컴파스로 작도가능 \(\iff\) \(n=2^k p_1 p_2 \cdots p_r\) (k ,r은 0이상의 정수, \(p_1, p_2, \cdots, p_r\) 은 서로 다른 페르마소수)
• 정다각형의 작도와 가우스와 정17각형의 작도 항목을 참조

역사

• 수학사 연표

관련된 항목들

• 정다각형의 작도
• 메르센 소수

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.
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).
3. Based on these results, one might conjecture (as did Fermat) that all Fermat numbers are prime.
4. To find the Fermat number F n for an integer n , you first find m = 2 n , and then calculate 2 m + 1.
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.
6. No fermat primes beyond (cid:8)4 have been found.
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.
8. + 1 is a Fermat number; such primes are called Fermat primes.
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.
10. The sum of the reciprocals of all the Fermat numbers (sequence A051158 OEIS) is irrational.
11. Indeed, the first five Fermat numbers F 0 , ..., F 4 are easily shown to be prime.
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.
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).
14. Now we know that all of the Fermat numbers are composite for the other n less than 31.
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.
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.
17. The sum of the reciprocals of all the Fermat numbers (sequence A051158 in OEIS) is irrational.
18. Indeed, the first five Fermat numbers F 0 ,...,F 4 are easily shown to be prime.
19. Indeed, even today, no other Fermat numbers are known to be prime!
20. Also show that every product of distinct Fermat numbers corresponds to a row of Pascal’s triangle mod 2.
21. Now, Gauss showed that we can construct a regular n-gon using straight-edge and compass if n is a prime Fermat number.
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.
23. There are two definitions of the Fermat number.
24. The much more commonly encountered Fermat numbers are a special case, given by the binomial number of the form .
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).
26. At present, however, only composite Fermat numbers are known for .
27. In other words, every prime of the form 2 n +1 is a Fermat number, and such primes are called Fermat primes.
28. From the last equation, we can deduce Goldbach's theorem: no two Fermat numbers share a common factor.
29. Indeed, the first five Fermat numbers F 0 ,..., F 4 are easily shown to be prime.
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").
31. A019434 List of Fermat primes: primes of form, for someIt is conjectured that there are only 5 terms.
32. Numbers of the form F n =22n+1 are now called Fermat numbers*, and when they’re prime, they’re called Fermat primes.
33. Fermat conjectured that all Fermat numbers are prime.
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.
35. So far, the only known Fermat primes are the ones that were known to Fermat.
36. Is there a formula or a way to know which of the Fermat numbers are prime?.
37. Prologue What are the known Fermat primes?
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.
39. We shall pronounce the last letter of Fermats name, as he did, when we include 2 among the Fermat primes, as he did.
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.
41. Fb;n = b2n + 1 and are particularly interesting since they have many characteristics of the heavily studied standard Fermat numbers Fn = F2;n.
42. The \Proth" program was created in 1997 to extend the search for large factors of Fermat numbers.
43. They are called Fermat numbers, named after the French mathematician Pierre de Fermat (1601 1665) who first studied numbers in this form.
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.
45. Primes in this form are called Fermat primes.
46. Up-to-date there are only five known Fermat primes.
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.
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.
49. The search for Mersenne and Fermat primes has been greatly extended since the 17th century.
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!
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.
52. Those are the only primes below 100,000 that I could show must be primitive for all but finitely many Fermat primes.
53. Fermat numbers are named after Pierre de Fermat.
54. Every prime of the form 2n + 1 is a Fermat number, and such primes are called Fermat primes.
55. Fermat numbers can be calculated recursively: To get the Nth number, multiply all Fermat numbers before it, and add two to the result.
56. Two previous posts (here and here) present an alternative proof that there are infinitely many prime numbers using the Fermat numbers.
57. Specifically, the proof is accomplished by pointing out that the prime factors of the Fermat numbers form an infinite set.
58. The numbers grow very rapidly since each Fermat number is obtained by raising 2 to a power of 2.
59. He demonstrated that the first 5 Fermat numbers , , , , are prime and conjectured that all Fermat numbers are prime.
60. Chapter 4 Fermat and Mersenne Primes 4.1 Fermat primes Theorem 4.1.
61. Fermat conjectured that the Fermat numbers are all prime.
62. We will come back to their solution shortly as we must first introduce the notion of a Fermat prime!
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.
64. So, he conjectured that all Fermat numbers are prime.
65. Funnily enough, it was shown by Leonhard Euler in 1732 (another famous mathematician), that only the next Fermat number is not prime.
66. It is still open whether there exist innitely many Fermat primes or innitely many composite Fermat numbers.
67. Especially, the formula for modulo Fermat primes is given. MSC2010: 11A07.
68. For the Fermat primes, i.e., the prime numbers of the form p 22k ` 1, we have Lppq 1. Proof.
69. We have veried it for k 0, 1, 2, 3, 4, the known Fermat primes.
70. As of 2021, the only known Fermat primes are F0 = 3, F1 = 5, F2 = 17, F3 = 257, and F4 = 65537.
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.
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.
73. We assume that m is divisible by at least one of the known Fermat prime numbers.
74. If m is divisible by one of the known Fermat primes, then m must be one of the following 11 Carmichael numbers.
75. Let 2 R R 22, there exists a generalized Fermat prime p = r2 2 be an integer.
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.
77. First, we encode integers to be multiplied as integers modulo generalized Fermat primes, and not as polynomials.
78. Section 4 studies generalized Fermat primes, and their relation to the Bateman-Horn conjecture.
79. 1. Introduction + 1 for n The Fermat numbers are given by Fn = 22n 0.
80. Notice that the rst ve Fermat numbers are prime, and it was initially conjectured (by Fermat) that all such num- bers are prime.
81. The sixth Fermat number is not prime, and no other Fermat primes are known.
82. An ecient test exists to determine whether or not a Fermat number is prime, called Pepins test.
83. This characterization uses uniquely values at most equal to tested Fermat number.
84. We actually are able to establish the direct implication which is a real important result in the primality tests for Fermat numbers.
85. Condition 2m + = pn requires p to be either a Mersenne or Fermat prime.

메타데이터

위키데이터

• ID : Q207264

Spacy 패턴 목록

• [{'LOWER': 'fermat'}, {'LEMMA': 'prime'}]
• [{'LOWER': 'fermat'}, {'LEMMA': 'number'}]