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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 . 1. Introduction and Historical Summary For a nonnegative integer n, the n-th Fermat number is Fn = 2 ... (Update)
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BibTeX entry: (Update)
R. P. Brent, Factorization of the tenth Fermat number, Math. Comp. 68 (1999), 429-451. http://citeseer.ist.psu.edu/brent98factorization.html More
@article{ brent99factorization,
author = "Richard P. Brent",
title = "Factorization of the tenth {Fermat} number",
journal = "Mathematics of Computation",
volume = "68",
number = "225",
pages = "429--451",
year = "1999",
url = "citeseer.ist.psu.edu/brent98factorization.html" }
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216 Volume 2: Seminumerical algorithms (context) - Knuth, of et al. - 1981
168 Algorithms for quantum computation: discrete logarithms and .. - Shor - 1994
134 Factoring integers with elliptic curves (context) - Lenstra - 1987
72 The development of the number field sieve (context) - Lenstra, Lenstra et al. - 1993
68 Prime numbers and computer methods for factorization (context) - Riesel - 1994
65 Speeding the Pollard and elliptic curve methods of factoriza.. (context) - Montgomery - 1987
57 Sequences of numbers generated by addition in formal groups .. (context) - Chudnovsky - 1986
55 Theorems in factorization and primality testing (context) - Pollard - 1974
51 a Fortran multiple-precision arithmetic package (context) - Brent, MP - 1978
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42 An improved Monte Carlo factorization algorithm (context) - Brent - 1980
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36 the frequency of numbers containing prime factors of a certa.. (context) - Dickman - 1930
34 Ordered cycle lengths in a random permutation (context) - Shepp, Lloyd - 1966
32 A Monte Carlo method for factorization (context) - Pollard - 1975
29 The factorization of the ninth Fermat number (context) - Lenstra, Lenstra et al. - 1993
27 Analysis of a simple factorization algorithm (context) - Knuth, Pardo - 1976
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20 Discrete weighted transforms and large-integer arithmetic (context) - Crandall, Fagin - 1994
17 the number of nonscalar multiplications necessary to evaluat.. (context) - Paterson, Stockmeyer - 1973
16 A practical analysis of the elliptic curve factoring algorit.. (context) - Silverman, Wagstaff - 1993
15 A survey of modern integer factorization algorithms - Montgomery - 1994
14 Parallel algorithms for integer factorisation (context) - Brent - 1990
14 Massively parallel elliptic curve factoring (context) - Dixon, Lenstra - 1993
13 Factorization of the eighth Fermat number (context) - Brent, Pollard - 1981
13 Topics in advanced scientific computation (context) - Crandall - 1996
13 a theory of factorization (context) - Shanks, number - 1971
13 Detecting perfect powers in essentially linear time - Bernstein
12 A tale of two sieves (context) - Pomerance - 1996
12 up to high powers (context) - Cunningham, Woodall et al. - 1925
11 Factorization of the tenth and eleventh Fermat numbers - Brent - 1996
11 Centrum voor Wiskunde en Informatica (context) - Brent, Riele et al. - 1992
11 Th'eorie des fonctions num'eriques simplement periodiques (context) - Lucas
11 On computing factors of cyclotomic polynomials - Brent - 1993
10 Courbes elliptiques et tests de primalit'e (context) - Morain - 1990
10 Large factors found by ECM (context) - Brent - 1995
10 Th'eorie des nombres (context) - Kraitchik - 1926
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9 An implementation of the number field sieve - Elkenbracht-Huizing - 1996
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7 New factors of Fermat numbers (context) - Gostin - 1995
7 Factor: an integer factorization program for the IBM PC (context) - Brent - 1989
7 Improved techniques for lower bounds for odd perfect numbers (context) - Brent, Cohen et al. - 1991
7 The number field sieve (context) - Pomerance - 1994
7 Factorization of the eleventh Fermat number (context) - Brent - 1989
6 An implementation of the elliptic curve integer factorizatio.. (context) - Bosma, Lenstra - 1995
6 Subtleties in the distribution of the numbers of points on e.. (context) - McKee
5 Informal preliminary report (context) - Suyama - 1985
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5 Mersenne and Fermat numbers (context) - Robinson - 1954
5 29--68 (context) - Atkin, Morain et al. - 1993
4 The asymptotic behaviour of a function occurring in the theo.. (context) - de Bruijn - 1951
4 A method of factorization and the factorization of (context) - Morrison, Brillhart - 1975
4 the numerical solution of a differential-difference equation.. (context) - Lune, Wattel - 1969
4 Succinct proofs of primality for the factors of some Fermat .. (context) - Brent - 1982
3 From Number Theory to Physics (context) - Cohen, curves - 1992
3 How was F 6 factored (context) - Williams - 1993
3 New Cullen primes (context) - Keller - 1995
3 Some miscellaneous factorizations (context) - Brillhart - 1963
3 personal communication by e-mail (context) - Montgomery - 1995
3 Tests for primality by the converse of Fermat's theorem (context) - Lehmer - 1927
3 Primality certificates for factors of some Fermat numbers (context) - Brent - 1995
2 Two new factors of Fermat numbers - Brent, Crandall et al. - 1997
1 Note sur la d'ecomposition du nombre (context) - Landry
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