## Factorization Of The Tenth Fermat Number (1999)

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Venue: | MATH. COMP |

Citations: | 22 - 10 self |

### BibTeX

@ARTICLE{Brent99factorizationof,

author = {Richard P. Brent},

title = {Factorization Of The Tenth Fermat Number},

journal = {MATH. COMP},

year = {1999},

volume = {68},

pages = {429--451}

}

### OpenURL

### Abstract

We describe the complete factorization of the tenth Fermat number F 10 by the elliptic curve method (ECM). F 10 is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The 40-digit factor was found after about 140 Mflop-years of computation. We also discuss the complete factorization of other Fermat numbers by ECM, and summarize the factorizations of F 5 ; : : : ; F 11 .

### Citations

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162 | Elliptic curves and primality proving
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Citation Context ...ound a 40-digit factor p40 = 4659775785220018543264560743076778192897 on October 20, 1995. The 252-digit cofactor c291/p40 = 13043 ···24577 was proved to be prime using the method of Atkin and Morain =-=[1]-=-. Later, a more elementary proof was found, using Selfridge’s “Cube Root Theorem” (see §9). Thus, the complete factorization is F10 = 45592577 ·6487031809 ·4659775785220018543264560743076778192897 ·p2... |

126 |
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Citation Context ....org/journal-terms-of-use432 R. P. BRENT 2. Variants of ECM The elliptic curve method (ECM) was discovered by H. W. Lenstra, Jr. [46] in 1985. Practical refinements were suggested by various authors =-=[8, 23, 50, 51, 72]-=-. We refer to [45, 52, 62, 71] for a general description of ECM, and to [24, 69] for relevant background. Suppose we attempt to find a factor of a composite number N, whichwecan assume not to be a per... |

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Citation Context ...llard [43, 61]. In 1990, Lenstra, Lenstra, Manasse and Pollard, with the assistance of many collaborators and approximately 700 workstations scattered around the world, completely factored F9 by SNFS =-=[44, 45]-=-. The factorization is F9 = 2424833 · 7455602825647884208337395736200454918783366342657 · p99 . In §8 we show that it would have been possible (though more expensive) to complete the factorization of ... |

51 | An improved monte carlo factorization algorithm - Brent - 1980 |

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Citation Context ...numbers F6,F7,... has been a challenge since Euler’s time. Because the Fn grow rapidly in size, a method which factors Fn may be inadequate for Fn+1. Historical details and references can be found in =-=[21, 35, 36, 44, 74]-=-, and some recent results are given in [17, 26, 27, 34]. In the following, pn denotes a prime number with n decimal digits (not necessarily the same at each occurrence). Similarly, cn denotes a compos... |

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Citation Context ....org/journal-terms-of-use432 R. P. BRENT 2. Variants of ECM The elliptic curve method (ECM) was discovered by H. W. Lenstra, Jr. [46] in 1985. Practical refinements were suggested by various authors =-=[8, 23, 50, 51, 72]-=-. We refer to [45, 52, 62, 71] for a general description of ECM, and to [24, 69] for relevant background. Suppose we attempt to find a factor of a composite number N, whichwecan assume not to be a per... |

45 |
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Citation Context ...a large random integer N with largest prime factor n1 and second-largest prime factor n2. Note that ρ(α) =µ(α, 1) and ρ2(α) =µ(α, α). The function ρ is usually called Dickman’s function after Dickman =-=[30]-=-, though some authors refer to Φ1 as Dickman’s function, and Vershik [73] calls ϕ1 =Φ′ 1 the Dickman-Goncharov function. It is known (see [8]) that ρ satisfies a differential-difference equation (12) ... |

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Citation Context ...2 Brillhart (p10) ECM 1995 Brent (p40,p252) 11 p6 · p ′ 6 · p21 · p22 · p564 Trial division 1899 Cunningham (p6,p′ 6 ) ECM 1988 Brent (p21,p22,p564) ECPP 1988 Morain (primality of p564) announcements =-=[9, 11]-=-, we describe the computation in §7. Further examples of factorizations obtained by ECM are given in §8. Rigorously proving primality of a number as large as the 564-digit factor of F11 is a nontrivia... |

39 |
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(Show Context)
Citation Context ...nt of the digital computer and more efficient algorithms. In 1970, Morrison and Brillhart [55] factored F7 = 59649589127497217 · p22 by the continued fraction method. Then, in 1980, Brent and Pollard =-=[18]-=- factored F8 = 1238926361552897 · p62 by a modification of Pollard’s “rho” method [6, 58]. The larger factor p62 of F8 was first proved prime by Williams [18, §4] using the method of Williams and Judd... |

29 |
Discrete weighted transforms and large-integer arithmetic
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Citation Context ...of the cost of ECM is in performing multiplications mod N. Our programs all use the classical O(w2 ) algorithm to multiply w-bit numbers. Karatsuba’s algorithm [38, §4.3.3] or other “fast” algorithms =-=[26, 28]-=- would be preferable for large w. The crossover point depends on details of the implementation. Morain [54, Ch. 5] states that Karatsuba’s method is worthwhile for w ≥ 800 on a 32-bit workstation. The... |

23 |
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Citation Context ...Because the Fn grow rapidly in size, a method which factors Fn may be inadequate for Fn+1. Historical details and references can be found in [21, 35, 36, 44, 74], and some recent results are given in =-=[17, 26, 27, 34]-=-. In the following, pn denotes a prime number with n decimal digits (not necessarily the same at each occurrence). Similarly, cn denotes a composite number with n decimal digits. Landry [41] factored ... |

19 |
Factorization of the eleventh Fermat number (preliminary report
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- 1989
(Show Context)
Citation Context ...e were confident that the quotient was indeed prime and that the factorization of F11 was complete. This was verified by Morain, as described in §9. The complete factorization of F11 was announced in =-=[9]-=-: F11 = 319489 · 974849 · 167988556341760475137 · 3560841906445833920513 · p564. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use446 R. P. BR... |

18 | Factorizations of b n \Sigma 1 for b - Brillhart, Lehmer, et al. - 1983 |

17 | Factorization of the tenth and eleventh Fermat numbers
- Brent
- 1997
(Show Context)
Citation Context ...ity proofs and “certificates” of primality for the factors of Fn, n ≤ 11. Attempts to factor Fermat numbers by ECM are continuing. For example, 27digit factors of F13 and F16 have recently been found =-=[13, 17]-=-. The smallest Fermat number which is not yet completely factored is F12. It is known that F12 has at least seven prime factors, and F12 = 114689 · 26017793 · 63766529 · 190274191361 · 125613213412556... |

17 |
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Citation Context ...s may apply to redistribution; see http://www.ams.org/journal-terms-of-use436 R. P. BRENT in [8] and has been implemented in several of our programs (see §5) and in the programs of A. Lenstra et al. =-=[4, 31, 45]-=-. There are several variations on the birthday paradox idea. We describe a version which is easy to implement and whose efficiency is comparable to that of the improved standard continuation. Followin... |

15 |
Large factors found by ECM
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Citation Context ...bution; see http://www.ams.org/journal-terms-of-use446 R. P. BRENT 8. Additional examples To show the capabilities of ECM, we give three further examples. Details and other examples are available in =-=[14]-=-. These examples do not necessarily illustrate typical behaviour of ECM. In December 1995, using program C with B1 = 370, 000, D = 255, e =6,we found the 40-digit factor p ′ 40 = 940985320569666416814... |

15 |
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Citation Context ...re not time-critical, such as input and output, are performed using the MP package [5]. Program C found the factorization of F11 (see §7) and many factors, of size up to 40 decimal digits, needed for =-=[16, 19]-=-. Keller [37] used program C to find factors up to 39 digits of Cullen numbers. D. A modification of MVFAC also runs on other machines with Fortran compilers, e.g., Sun 4 workstations. For machines us... |

13 | Factorization of cyclotomic polynomials
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(Show Context)
Citation Context ...enerate tables [19], taking into account License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use442 R. P. BRENT algebraic and Aurifeuillian factors =-=[12]-=-, and accessing a database of over 230, 000 known factors. As a byproduct, program B can produce lists of composites which are used as input to other programs. C. When a vector processor 1 became avai... |

13 | An implementation of the number field sieve
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Citation Context ...st, the running times of the MPQS and GNFS methods can be predicted fairly well, because they depend mainly on the size of the number being factored, and not on the size of the (unknown) factors, see =-=[3, 32, 60]-=-. An important question is how long to spend on ECM before switching to a more predictable method such as MPQS/GNFS. This question is considered in [71, §7], but our approach is different. Theorem 3 o... |

13 |
Trabb Pardo, Analysis of a simple factorization algorithm
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(Show Context)
Citation Context ...rformance of ECM In order to predict the performance of ECM, we need some results on the distribution of prime factors of random integers. These results and references to earlier work may be found in =-=[39, 73]-=-. 4.1. Prime factors of random integers. Let n1(N) ≥ n2(N) ≥ ... be the prime factors of a positive integer N. Thenj(N) are not necessarily distinct. For convenience we take nj(N) =1ifNhaslessthanjpri... |

12 | Algorithm 524: MP, a Fortran multiple-precision arithmetic package
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Citation Context ...ions (flops). INT and DFLOAT operations are used to split a product into high and low-order parts. Operations which are not time-critical, such as input and output, are performed using the MP package =-=[5]-=-. Program C found the factorization of F11 (see §7) and many factors, of size up to 40 decimal digits, needed for [16, 19]. Keller [37] used program C to find factors up to 39 digits of Cullen numbers... |

11 |
te Riele, Factoring integers with large prime variations of the quadratic sieve
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Citation Context ...st, the running times of the MPQS and GNFS methods can be predicted fairly well, because they depend mainly on the size of the number being factored, and not on the size of the (unknown) factors, see =-=[3, 32, 60]-=-. An important question is how long to spend on ECM before switching to a more predictable method such as MPQS/GNFS. This question is considered in [71, §7], but our approach is different. Theorem 3 o... |

10 | Projects in scientific computation - Crandall - 1994 |

8 |
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Citation Context ...s may apply to redistribution; see http://www.ams.org/journal-terms-of-use436 R. P. BRENT in [8] and has been implemented in several of our programs (see §5) and in the programs of A. Lenstra et al. =-=[4, 31, 45]-=-. There are several variations on the birthday paradox idea. We describe a version which is easy to implement and whose efficiency is comparable to that of the improved standard continuation. Followin... |

8 | The twenty-second Fermat number is composite
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- 1995
(Show Context)
Citation Context ...Because the Fn grow rapidly in size, a method which factors Fn may be inadequate for Fn+1. Historical details and references can be found in [21, 35, 36, 44, 74], and some recent results are given in =-=[17, 26, 27, 34]-=-. In the following, pn denotes a prime number with n decimal digits (not necessarily the same at each occurrence). Similarly, cn denotes a composite number with n decimal digits. Landry [41] factored ... |

8 |
Note sur la décomposition du nombre 2 64
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Citation Context ...7, 26, 27, 34]. In the following, pn denotes a prime number with n decimal digits (not necessarily the same at each occurrence). Similarly, cn denotes a composite number with n decimal digits. Landry =-=[41]-=- factored F6 = 274177 · p14 in 1880, but significant further progress was only possible with the development of the digital computer and more efficient algorithms. In 1970, Morrison and Brillhart [55]... |

6 |
de Bruijn, The asymptotic behaviour of a function occurring in the theory of primes
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Citation Context ...β → 1+. The results (12)–(14) follow from Vershik’s general result (10), although it is possible to derive them more directly, as in [8, 38, 39]. Sharp asymptotic results for ρ are given by de Bruijn =-=[22, 48]-=-, and an asymptotic result for µ is stated in [8]. To predict the performance of phase 1 of ECM it is enough to know that (15) as α →∞. ρ(α − 1) ρ(α) ∼−log ρ(α) ∼ α log α 4.2. Heuristic analysis of ph... |

6 | Factorisation of y n \Upsilon 1, y - Cunningham, Woodall - 1925 |

6 |
Factors of Fermat Numbers and Large Primes of the Form k2 n + 1
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Citation Context ...numbers F6,F7,... has been a challenge since Euler’s time. Because the Fn grow rapidly in size, a method which factors Fn may be inadequate for Fn+1. Historical details and references can be found in =-=[21, 35, 36, 44, 74]-=-, and some recent results are given in [17, 26, 27, 34]. In the following, pn denotes a prime number with n decimal digits (not necessarily the same at each occurrence). Similarly, cn denotes a compos... |

6 | Tests for primality by the converse of fermat’s theorem - Lehmer - 1927 |

5 |
Factor: an integer factorization program for the IBM
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Citation Context ... + K2B3). Program A ran on various machines, including Sun 3 and VAX, and found many factors of up to 25 decimal digits [19]. B. A simple Turbo Pascal implementation was written in 1986 for an IBM PC =-=[10]-=-. The implementation of multiple-precision arithmetic is simple but inefficient. Program B is mainly used to generate tables [19], taking into account License or copyright restrictions may apply to re... |

5 | Two new factors of Fermat numbers
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(Show Context)
Citation Context ...Because the Fn grow rapidly in size, a method which factors Fn may be inadequate for Fn+1. Historical details and references can be found in [21, 35, 36, 44, 74], and some recent results are given in =-=[17, 26, 27, 34]-=-. In the following, pn denotes a prime number with n decimal digits (not necessarily the same at each occurrence). Similarly, cn denotes a composite number with n decimal digits. Landry [41] factored ... |

5 | te Riele, Factorizations of a n \Sigma 1, 13 a - Brent, J - 1992 |

5 |
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(Show Context)
Citation Context ...numbers F6,F7,... has been a challenge since Euler’s time. Because the Fn grow rapidly in size, a method which factors Fn may be inadequate for Fn+1. Historical details and references can be found in =-=[21, 35, 36, 44, 74]-=-, and some recent results are given in [17, 26, 27, 34]. In the following, pn denotes a prime number with n decimal digits (not necessarily the same at each occurrence). Similarly, cn denotes a compos... |

4 |
Elliptic curves, From Number Theory to Physics (edited by
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(Show Context)
Citation Context ...) was discovered by H. W. Lenstra, Jr. [46] in 1985. Practical refinements were suggested by various authors [8, 23, 50, 51, 72]. We refer to [45, 52, 62, 71] for a general description of ECM, and to =-=[24, 69]-=- for relevant background. Suppose we attempt to find a factor of a composite number N, whichwecan assume not to be a perfect power [2], [44, §2.5]. Let p be the smallest prime factor of N. In practice... |

4 |
New factors of Fermat numbers
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(Show Context)
Citation Context |

4 |
te Riele, Factorizations of a n
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(Show Context)
Citation Context ...at, as in §2.2–§3.3, the number of multiplications mod N per curve is about K1B1 + K2B3). Program A ran on various machines, including Sun 3 and VAX, and found many factors of up to 25 decimal digits =-=[19]-=-. B. A simple Turbo Pascal implementation was written in 1986 for an IBM PC [10]. The implementation of multiple-precision arithmetic is simple but inefficient. Program B is mainly used to generate ta... |

3 |
Succinct proofs of primality for the factors of some Fermat numbers
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(Show Context)
Citation Context ...er value.) The“lucky”curveisdefinedasin§2.3with σ = 862263446. The group order is 2 2 · 3 2 · 5 2 · 7 · 331 · 1231 · 1289 · 6277 · 68147 · 1296877 ·9304783 ·9859051 · 44275577. 9. Primality proofs In =-=[7]-=- we gave primality certificates for the prime factors of F5,... ,F8,using the elegant method pioneered by Lucas [47, p. 302], Kraitchik [40, p. 135] and Lehmer [42, p. 330]. To prove p prime by this m... |

2 |
Primality certificates for factors of some Fermat numbers
- Brent
- 1995
(Show Context)
Citation Context ...ion V3.4.1 of ecpp, running on a 60 Mhz SuperSparc, established the primality of p564 in 28 hours. It took only one hour to prove p252 prime by the same method. Primality “certificates” are available =-=[15]-=-. They can be checked using a separate program xcheckcertif. 10. When to use ECM, and prospects for F12 When factoring large integers by ECM we do not usually know the size of the factors in advance. ... |

2 |
Some miscellaneous factorizations
- Brillhart
- 1963
(Show Context)
Citation Context ...te in 1952 by Robinson [64], using Pépin’s test on the SWAC. A small factor, 45592577, was found by Selfridge [65] in 1953 (also on the SWAC). Another small factor, 6487031809, was found by Brillhart =-=[20]-=- in 1962 on an IBM 704. Brillhart later found that the cofactor was a 291-digit composite. Thus, it was known that F10 = 45592577 · 6487031809 · c291. This paper describes the complete factorization o... |

2 |
On the field of combinatory analysis, Izv
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(Show Context)
Citation Context ...1 Φk = 0 θk ∫ α1 ··· θ2 Φ1 ( θk 1−θ1 −···−θk ) dθ1 ...dθk−1dθk θ1 ...θk−1θk Knuth and Trabb Pardo [39] observe an interesting connection with the distribution of cycle lengths in a random permutation =-=[33, 67]-=-. In fact, the distribution of the number of digits of the j-th largest prime factors of n-digit random integers is asymptotically the same as the distribution of lengths of the j-th longest cycles in... |

2 | Théorie des nombres, Tome 2 - Kraitchik - 1926 |

1 | Note sur la d'ecomposition du nombre 2 +1 - Landry |

1 |
Also updates to Update 2.9
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Citation Context |

1 |
Factorisation of y n ∓ 1, y =2,3,5,6,7,10
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(Show Context)
Citation Context ...e the factorization of F9 and F10. Infact, F11 = 319489 · 974849 · 167988556341760475137 · 3560841906445833920513 · p564, where p564 = 17346 ···34177. The two 6-digit factors were found by Cunningham =-=[21, 29]-=- in 1899, and remaining factors were found by the present author in May 1988. The reason why F11 could be completely factored before F9 and F10 is that the difficulty of completely factoring numbers b... |