## Three New Factors of Fermat Numbers (2000)

### Cached

### Download Links

- [web.comlab.ox.ac.uk]
- [wwwmaths.anu.edu.au]
- [www.ams.org]
- [www.ams.org]
- [ftp.comlab.ox.ac.uk]
- [people.reed.edu]
- [www.ams.org]
- DBLP

### Other Repositories/Bibliography

Venue: | Math. Comp |

Citations: | 4 - 0 self |

### BibTeX

@ARTICLE{Brent00threenew,

author = {R. P. Brent and R. E. Crandall and K. Dilcher and C. Van Halewyn},

title = {Three New Factors of Fermat Numbers},

journal = {Math. Comp},

year = {2000}

}

### OpenURL

### Abstract

We report the discovery of a new factor for each of the Fermat numbers F 13 ,F 15 ,F 16 . These new factors have 27, 33 and 27 decimal digits respectively. Each factor was found by the elliptic curve method. After division by the new factors and previously known factors, the remaining cofactors are seen to be composite numbers with 2391, 9808 and 19694 decimal digits respectively. 1.

### Citations

675 | The Art of Computer Programming, volume 2: Seminumerical Algorithms - Knuth - 1988 |

181 |
Algebraic Complexity Theory
- Bürgisser, Claussen, et al.
- 1997
(Show Context)
Citation Context ...st as large as 50B1. 4.3. GCD computation. It is nontrivial to compute GCDs for numbers in the F21 region. We used a recursive GCD implementation by J. P. Buhler [6], based on the Schönhage algorithm =-=[7, 28]-=-. The basic idea is to recursively compute a 2 × 2 matrix M such that if v =(a, b) T is the column vector containing the two numbers whose GCD we desire, then Mv =(0, gcd(a, b)) T . The matrix M is a ... |

102 | Sequences of numbers generated by addition in formal groups and new primality and factoring tests - Chudnovsky, Chudnovsky - 1987 |

52 | Factoring by electronic mail
- Lenstra, Manasse
(Show Context)
Citation Context ...vision [16, 18]. 2. The elliptic curve method ECM was invented by H. W. Lenstra, Jr. [23]. Various practical refinements were suggested by Brent [1], Montgomery [24, 25], and Suyama [32]. We refer to =-=[3, 14, 22, 26, 31]-=- for a description of ECM and some of its implementations. In the following, we assume that ECM is used to find a prime factor p>3ofa composite number N, not a prime power [21, §2.5]. The first-phase ... |

49 | The factorization of the ninth Fermat number - Lenstra, Jr, et al. - 1993 |

46 | Some integer factorization algorithms using elliptic curves
- Brent
- 1986
(Show Context)
Citation Context ...ors of larger numbers are customarily found by trial division [16, 18]. 2. The elliptic curve method ECM was invented by H. W. Lenstra, Jr. [23]. Various practical refinements were suggested by Brent =-=[1]-=-, Montgomery [24, 25], and Suyama [32]. We refer to [3, 14, 22, 26, 31] for a description of ECM and some of its implementations. In the following, we assume that ECM is used to find a prime factor p>... |

29 |
Discrete weighted transforms and large-integer arithmetic
- Crandall, Fagin
- 1994
(Show Context)
Citation Context ...st of ECM is in performing multiplications mod N. Our Cruncher programs all use the classical O(w2 ) algorithm to multiply w-bit numbers. Karatsuba’s algorithm [19, §4.3.3] or other “fast” algorithms =-=[11, 13]-=- are preferable for large w on a workstation. The crossover point depends on details of the implementation. Morain [27, Ch. 5] states that Karatsuba’s method is worthwhile for w ≥ 800 on a 32-bit work... |

23 | Topics in Advanced Scientific Computation - Crandall - 1996 |

22 | Factorization of the tenth fermat number
- Brent
- 1999
(Show Context)
Citation Context ... 23. For a brief history of attempts to factor Fermat numbers, we refer to [3, §1] and [5]. In recent years several factors of Fermat numbers have been found by the elliptic curve method (ECM). Brent =-=[2, 3, 4]-=- completed the factorization of F10 (by finding a 40-digit factor) and F11. He also “rediscovered” the 49-digit factor of F9 and the five known prime factors of F12. Crandall [10] discovered two 19-di... |

19 |
Factorization of the eleventh Fermat number (preliminary report
- Brent
- 1989
(Show Context)
Citation Context ... 23. For a brief history of attempts to factor Fermat numbers, we refer to [3, §1] and [5]. In recent years several factors of Fermat numbers have been found by the elliptic curve method (ECM). Brent =-=[2, 3, 4]-=- completed the factorization of F10 (by finding a 40-digit factor) and F11. He also “rediscovered” the 49-digit factor of F9 and the five known prime factors of F12. Crandall [10] discovered two 19-di... |

17 | Factorization of the tenth and eleventh Fermat numbers
- Brent
- 1997
(Show Context)
Citation Context ... 23. For a brief history of attempts to factor Fermat numbers, we refer to [3, §1] and [5]. In recent years several factors of Fermat numbers have been found by the elliptic curve method (ECM). Brent =-=[2, 3, 4]-=- completed the factorization of F10 (by finding a 40-digit factor) and F11. He also “rediscovered” the 49-digit factor of F9 and the five known prime factors of F12. Crandall [10] discovered two 19-di... |

17 |
Massively parallel elliptic curve factoring
- Dixon, Lenstra
- 1993
(Show Context)
Citation Context ...vision [16, 18]. 2. The elliptic curve method ECM was invented by H. W. Lenstra, Jr. [23]. Various practical refinements were suggested by Brent [1], Montgomery [24, 25], and Suyama [32]. We refer to =-=[3, 14, 22, 26, 31]-=- for a description of ECM and some of its implementations. In the following, we assume that ECM is used to find a prime factor p>3ofa composite number N, not a prime power [21, §2.5]. The first-phase ... |

14 |
The Dubner PC Cruncher { a microcomputer coprocessor card for doing integer arithmetic, review in
- Caldwell
- 1993
(Show Context)
Citation Context ...ake (x1 ::z1). It is not necessary to specify b or y1. When using (2) we assume that the starting point is chosen as in (3), with σ a pseudo-random integer. 3. The Dubner Cruncher The Dubner Cruncher =-=[8, 15]-=- is a board which plugs into an IBM-compatible PC. The board has a digital signal processing chip (LSI Logic L64240 MFIR) which, when used for multiple-precision integer arithmetic, can multiply two 5... |

10 |
Projects in scientific computation
- Crandall
- 1994
(Show Context)
Citation Context ...od (ECM). Brent [2, 3, 4] completed the factorization of F10 (by finding a 40-digit factor) and F11. He also “rediscovered” the 49-digit factor of F9 and the five known prime factors of F12. Crandall =-=[10]-=- discovered two 19-digit factors of F13. This paper reports the discovery of 27-digit factors of F13 and F16 (the factor of F13 was announced in [3, §8]) and of a 33-digit factor of F15. All three fac... |

8 | The twenty-second Fermat number is composite
- Crandall, Doenias, et al.
- 1995
(Show Context)
Citation Context ...remaining cofactors are seen to be composite numbers with 2391, 9808 and 19694 decimal digits respectively. 1. Introduction For a nonnegative integer n, then-th Fermat number is Fn =22n+1. Itis known =-=[12]-=- that Fn is prime for 0 ≤ n ≤ 4, and composite for 5 ≤ n ≤ 23. For a brief history of attempts to factor Fermat numbers, we refer to [3, §1] and [5]. In recent years several factors of Fermat numbers ... |

6 |
Factors of Fermat Numbers and Large Primes of the Form k2 n + 1
- Keller
- 1983
(Show Context)
Citation Context ...nd F15 are given in §5, §6 and§7respectively. F16 is probably the largest number for which a nontrivial factor has been found by ECM. Factors of larger numbers are customarily found by trial division =-=[16, 18]-=-. 2. The elliptic curve method ECM was invented by H. W. Lenstra, Jr. [23]. Various practical refinements were suggested by Brent [1], Montgomery [24, 25], and Suyama [32]. We refer to [3, 14, 22, 26,... |

5 |
Two new factors of Fermat numbers
- Hallyburton, Brillhart
- 1975
(Show Context)
Citation Context ...nutes for phase 2). At the time three prime factors of F13 were known: F13 = 2710954639361 · 2663848877152141313 · 3603109844542291969 · c2417. The first factor was found by Hallyburton and Brillhart =-=[17]-=-. The second and third factors were found by Crandall [10] on Zilla net (a network of about 100 workstations) in January and May 1991, using ECM. On June 16, 1995 our Cruncher program found a fourth f... |

4 |
New factors of Fermat numbers
- Gostin
- 1995
(Show Context)
Citation Context ...nd F15 are given in §5, §6 and§7respectively. F16 is probably the largest number for which a nontrivial factor has been found by ECM. Factors of larger numbers are customarily found by trial division =-=[16, 18]-=-. 2. The elliptic curve method ECM was invented by H. W. Lenstra, Jr. [23]. Various practical refinements were suggested by Brent [1], Montgomery [24, 25], and Suyama [32]. We refer to [3, 14, 22, 26,... |

3 |
The Dubner PC Cruncher: Programmers Guide and Function Reference
- Dubner, Dubner
- 1993
(Show Context)
Citation Context ...ake (x1 ::z1). It is not necessary to specify b or y1. When using (2) we assume that the starting point is chosen as in (3), with σ a pseudo-random integer. 3. The Dubner Cruncher The Dubner Cruncher =-=[8, 15]-=- is a board which plugs into an IBM-compatible PC. The board has a digital signal processing chip (LSI Logic L64240 MFIR) which, when used for multiple-precision integer arithmetic, can multiply two 5... |

2 |
personal communication to Crandall
- Buhler
- 1993
(Show Context)
Citation Context ...e choices of stage two limit B2 at least as large as 50B1. 4.3. GCD computation. It is nontrivial to compute GCDs for numbers in the F21 region. We used a recursive GCD implementation by J. P. Buhler =-=[6]-=-, based on the Schönhage algorithm [7, 28]. The basic idea is to recursively compute a 2 × 2 matrix M such that if v =(a, b) T is the column vector containing the two numbers whose GCD we desire, then... |

1 |
On the factorization of 2
- Kraitchik
(Show Context)
Citation Context ...15 was attempted during the Spring and early Summer of 1997. It was known that F15 = 1214251009 · 2327042503868417 · c9840, where the 13- and 16-digit prime factors were found by Kraitchik (1925; see =-=[20]-=-) and Gostin (1987; see [16]) respectively. On July 3, 1997 we found the new factor p33 = 168768817029516972383024127016961 after running only three curves with B1 =10 7 and B2 =50B1. Each curve took ... |