J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagstaff, Jr., Factorizations of b \Sigma 1, b = 2; 3; 5; 6; 7; 10; 11; 12 up to high powers, 2nd ed., Amer. Math. Soc., Providence, RI,  Home/Search   Document Not in Database   Summary   Related Articles   Check

....take about 3800 times as long and require about 62 times as much space, or about 3.3Gbytes per sieve processor. If one tries to use an out of core siever, it slows the computations down considerably. 7. Numerical Results The author has factored hundreds of numbers from the Cunningham project [7] and from its extensions [4] This section gives suggested choices, as a function of the size of the number being factored, for the algorithm parameters. These have been found to work well on Pentium based processors. For other processors, with di#erent cache sizes and instruction sets, these ....

J. Brillhart, D.H. Lehmer, J. Selfridge, B. Tuckerman, & S.S. Wagsta# Jr., Factorizations of b for b = 2, 3, 5, 6, 7, 10, 11, 12 up to High Powers, Contemporary Mathematics #23, Amer. Math. Soc., 1987.

....and 19694 decimal digits respectively. 1. Introduction For a nonnegative integer n,then th Fermat number is Fn =2 2 n 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 have been found by the elliptic curve method (ECM) Brent [2, 3, 4] completed the factorization of F 10 (by finding a 40 digit factor) and F 11 . He also rediscovered the 49 digit factor of F 9 and the five known prime factors of F 12 . ....

J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagsta#, Jr., Fac t orizations of b n 1, b =2, 3, 5, 6, 7, 10, 11, 12 up to high powers, 2nd ed., Amer. Math. Soc., Providence, RI, 1988. MR 90d:11009

....19694 decimal digits respectively. 1. Introduction For a nonnegative integer n, the n th Fermat number is F n = 2 2 n 1. It is known [12] that F n 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, x1] 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 F 10 (by nding a 40 digit factor) and F 11 . He also rediscovered the 49 digit factor of F 9 and the ve known prime factors of F 12 . Crandall ....

J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagsta, Jr., Factorizations of b n  1, b = 2; 3; 5; 6; 7; 10; 11; 12 up to high powers, 2nd ed., Amer. Math. Soc., Providence, RI, 1988.

....9808 and 19694 decimal digits respectively. 1. Introduction For a nonnegative integer n, the n th Fermat number is Fn = 2 2 n 1. It is 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, x1] 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 F 10 (by finding a 40 digit factor) and F 11 . He also rediscovered the 49 digit factor of F 9 and the five known prime factors of F 12 . ....

J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagstaff, Jr., Factorizations of b n \Sigma 1, b = 2; 3; 5; 6; 7; 10; 11; 12 up to high powers, 2nd ed., Amer. Math. Soc., Providence, RI, 1988.

....purposes one need not compute the entire P p2f.b. E(p) but instead it will generally be sufficient to sum up to some fixed bound, like 10,000. 3. 8 The factorization of a 116 digit number We have used siqs to be the first to factor about a dozen numbers for the Cunningham project [3]. The largest was a 116 digit divisor of 2 481 2 241 1, completed on June 15, 1996. This number was on the Most Wanted List for the project. The number we factored was (on the following two lines) 9535455903208375257494587744656958072003190386397547282545n ....

J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, S. S. Wagstaff, Jr., Factorizations of b n \Sigma 1, b=2,3,5,6,7,10,11,12 up to high powers, second edition, Contemporary Mathematics, vol. 22, Providence: A.M.S., 1988.

....provides useful examples for several topics in elementary number theory. As the theory of factoring has developed, such numbers have proved to be fertile ground for the initial development of factoring techniques, which may subsequently be generalizable to factoring arbitrary integers. The book [BLS] contains a good historical account up to the time it was written; and for up to date data see the website http: www.cerias.purdue.edu homes ssw cun index.html. Indeed even the number field sieve, the latest general factoring technique, was first suggested by Pollard to attack numbers in the ....

....If nN is four times an odd integer and r is twice an odd integer. 4 ANDREW GRANVILLE AND PETER PLEASANTS In each case this last factorization is given by taking x = rT 2 , with T = b(t c) q=2 , in Schinzel s identities (2) and obtaining the difference of two squares. Among the data in [BLS] on numbers factored by other methods there are examples which suggest that there may be some other way to extend Aurifeuillian factorizations. For instance, Wagstaff points out the following interesting example from the Cunningham project: 6 106 1 6 2 1 = 26713 Theta ....

J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman and S. S. Wagstaff Jr, Factorizations of b n \Sigma 1; b = 2; 3; 5; 6; 7; 10; 11; 12 up to high powers, Amer. Math. Soc., Providence, RI, 1988.

....at the Electrical Engineering and Computer Science Department, University of Wisconsin Milwaukee, U.S.A. yy The author is with the School of Information Science, Japan Advanced Institute of Science and Technology, Ishikawa ken, 923 12 Japan. grams to factor integers for the Cunningham Project [3]. 2. The Quadratic Sieve algorithm The QS algorithm is currently the fastest algorithm for factoring numbers in the 100 digit range and with no small prime factors. The general technique (of QS as well as of most modern factoring algorithms) can be traced to ideas published in 1926 by Kraitchik ....

Brillhart, J., Lehmer, D. H., Selfridge, J. L., Tuckerman, B., Wagstaff, S. S. Jr.: Factorizations of b n \Sigma 1, b=2,3,5,6,7,10,11,12 up to high powers. In Contemporary PERALTA and OKAMOTO: FACTORING HARD INTEGERS ON A PARALLEL MACHINE 5 Mathematics vol. 22. American Mathematical Society 1988.

.... means of a general purpose primality testing algorithm such as that described in [1, 11, 10, 6] or the Elliptic Curve Primality Proving (ECPP) algorithm due to Atkin [2, 25, 28] For a survey of primality testing, see [18] Another point is to certify large primes, such as the Cunningham numbers [8], which sometimes have more than 400 digits. The purpose of this paper is to explain how the ECPP algorithm has been implemented on a network of workstations and used to test some numbers with more than 500 digits for primality. In particular, it is now routine to test 800 digit numbers and it is ....

....i as described in [26, 27] 2. find an equation of the curve E i whose invariant is j and cardinality m i ; 3. verify the condition of theorem (2) end. For more details, the reader is referred to [2] 3 Large primes The author used ECPP to test about fifty numbers from the Cunningham tables [8] and some others, namely S p = 1 p 2) p (1 Gamma p 2) p ) 2 for p 2 f1493; 1901g with respectively 572 and 728 digits, in 30 and 40 days on a single SUN 3 60. Indeed, a simple extrapolation shows that testing a 1000 digit number would require about 6 months (at least) We must do ....

J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagstaff, Jr. Factorizations of b n \Sigma 1; b = 2; 3; 5; 6; 7; 10; 11; 12 up to high powers, 2 ed. No. 22 in Contemporary Mathematics. AMS, 1988.

....Quadratic Sieve (MPQS) and its refinements have been the fastest general purpose factorization algorithms in practice. In this paper we describe the results obtained from a new refinement known as the Hypercube variation. We used our programs to factor integers for the Cunningham Project [3]. We also show how to predict the number of semi smooth residues (see section 2.2) and the number of matches for the single large prime variation of the algorithm. Using our methods, predicting optimal parameters becomes much easier. 2 The Quadratic Sieve algorithm The QS algorithm is currently ....

Brillhart, J., Lehmer, D. H., Selfridge, J., Tuckerman, B., S.S. Wagstaff, J.: Factorizations of b n \Sigma 1, b=2,3,5,6,7,10,11,12 up to high powers. In Contemporary Mathematics vol. 22. American Mathematical Society 1988.

....project I implemented Atkin s algorithm on a SUN 3 60 using the language Le Lisp 15.21 developed in INRIA. This language can handle arbitrarily large integers and the basic arithmetic routines are written in assembly. With that implementation, I was able to prove the primality of 43 numbers of [21]. Those numbers are listed below. d name d name 222 2; 1958M 284 2; 2338M 228 2; 1594M 284 2; 1096 228 2; 1874M 286 2; 2102M 236 2; 808 288 2; 1049 Gamma 237 2; 979 Gamma 294 2; 2126L 237 2; 883 296 2; 2122L 237 2; 1886L 301 2; 1061 245 2; 844 307 2; 2242M 255 2; 2366M 312 2; 1189 260 2; ....

J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman. Factorizations of b n \Sigma 1; b = 2; 3; 5; 6; 7; 10; 11; 12 up to high powers. Contemporary Mathematics, 22, AMS, 1983.

....is not prime, provided it has a large prime factor ( 10 40 , preferably, for reasons discussed above) Table 6 presents a selection of values of n between 127 and 521 for which the complete factorization of 2 n 1 is known and includes a very large prime factor. This table is drawn from [14], except that the primality of the 105 digit factor of 2 373 1 was proved by the author using the CohenLenstra [17] version of the Adleman Pomerance Rumely primality test [2] Also included are the two values n = 881 and n = 1063, for which the cofactors have not been shown to be prime, ....

J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagstaff, Jr., Factorizations of b n 1, b = 2 , 3 , 5 , 6 , 7 , 10 , 11 , 12 up to High Powers, Am. Math. Society, 1983.

....Polynomial Quadratic Sieve (MPQS) and its refinements have been the fastest general purpose factorization algorithms in practice. In this paper we describe the implementation of a new refinement known as the Hypercube variation [8] We used our program to factor integers for the Cunningham Project [3]. We also show how to predict the number of semi smooth residues (see section 2.2) and the number of matches for the single large prime variation of the algorithm. Using our methods, predicting optimal parameters becomes much easier. In section 2 we briefly explain HMPQS and how it evolved. For a ....

Brillhart, J., Lehmer, D. H., Selfridge, J., Tuckerman, B., S. Wagstaff, J.: Factorizations of b n \Sigma 1, b=2,3,5,6,7,10,11,12 up to high powers. In Contemporary Mathematics vol. 22. American Mathematical Society 1988.

....of curves defined over Q, for each of which the group of rational points contains a subgroup isomorphic to Z=2Z Theta Z=8Z, and a computable rational point of infinite order. We also discuss the limited use of curves with complex multiplication in the special contexts of the Cunningham project [2] and primality proving [1] 1.2 In order to construct curves with prescribed factors of k, we can look at curves defined over Q which have large torsion groups. By a theorem of Mazur [6] we know that the only possible torsion groups over Q are E tor (Q) ae Z=mZ; m = 1; 2; 10 or 12; ....

....So on balance it seems that we gain nothing in the general case. However, there are two cases where it can be guaranteed that all the prime factors of our composite N will be norms in some quadratic field Q( p GammaD) First, in the slightly artificial environment of the Cunningham Project [2], we may have many square roots and discriminants available. For example, if N is the result of removing the algebraic factors from 7 660 1, then all primes p dividing N have p = 1320n 1 with (7=p) Gamma1) n . Thus we can use Q( p GammaD) for D = 3; 4; 8; 11; 15; 20; 24; 40; 88; ....

J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagstaff, Jr. Factorizations of b n \Sigma 1; b = 2; 3; 5; 6; 7; 10; 11; 12 up to high powers, 2 ed. No. 22 in Contemporary Mathematics. AMS, 1988.

....the factorization of Weber polynomials over their genus field. In Section 8, we detail the computational routines we use in the implementation: Section 9 contains some typical running times for numbers of less than 300 digits and also some running times for larger numbers, most of all taken from [18] or discovered by the authors. Section 10 is briefly concerned with the second problem mentioned above, namely that of the actual proof we get by ECPP. Notation. Throughout the paper, N will denote a probable prime, which means that N was not declared composite by any of the probabilistic ....

....with all D with h less than 51 and some others (see Section 8.2.3) The order of magnitued of the time needed is given in equivalent time for a SUN 3 60. 9. 2 Some large primes Both authors used their implementations to give primality proofs for the probable primes of the Cunningham Tables [18]. The first author did some with 212 to 343 digits (namely the cofactor of 2; 1171 ) 18, Update# 5] and the second author completed the long standing list (about 50 numbers with more than 200 digits) The second author verified the primality of the cofactor of F 11 (564 digits) 18, Update 2.2] ....

[Article contains additional citation context not shown here]

J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagstaff, Jr. Factorizations of b n \Sigma 1; b = 2; 3; 5; 6; 7; 10; 11; 12 up to high powers, 2 ed. No. 22 in Contemporary Mathematics. AMS, 1988.

....the factorization of Weber polynomials over their genus field. In Section 8, we detail the computational routines we use in the implementation: Section 9 contains some typical running times for numbers of less than 300 digits and also some running times for larger numbers, most of all taken from [19] or discovered by the authors. Section 10 is briefly concerned with the second problem mentioned above, namely that of the actual proof we get by ECPP. Notation. Throughout the paper, N will denote a probable prime, which means that N was not declared composite by any of the probabilistic ....

....with all D with h less than 51 and some others (see Section 8.2.3) The order of magnitude of the time needed is given in equivalent time for a DecStation 5000. 9. 2 Some large primes Both authors used their implementations to give primality proofs for the probable primes of the Cunningham Tables [19]. The first author did some with 212 to 343 digits (namely the cofactor of 2; 1171 ) 19, Update# 5] and the second author completed the long standing list (about 50 numbers with more than 200 digits) The second author verified the primality of the cofactor of F 11 (564 digits) 19, Update 2.2] ....

[Article contains additional citation context not shown here]

J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagstaff, Jr. Factorizations of b n \Sigma 1; b = 2; 3; 5; 6; 7; 10; 11; 12 up to high powers, 2 ed. No. 22 in Contemporary Mathematics. AMS, 1988.

....with 2391 and 19694 decimal digits respectively. 1. Introduction For a nonnegative integer n, the n th Fermat number is Fn = 2 2 n 1. It is 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, x1] 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 F 10 (by finding a 40 digit factor) and F 11 . He also rediscovered the 49 digit factor of F 9 and the five known prime factors of F 12 . ....

J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagstaff, Jr., Factorizations of b n \Sigma 1, b = 2; 3; 5; 6; 7; 10; 11; 12 up to high powers, 2nd ed., Amer. Math. Soc., Providence, RI, 1988.

....that 4#(E(Z=pZ) x Gamma 2) 2 Dy 2 . Moreover, this curve can be computed explicitly using any root of the polynomial HD (X) modulo p. The algorithm then proceeds as in the classical DOWNRUN process of the well known primality proving algorithms based on the converse of Fermat s Theorem [6, 22]. 4 Building curves of given order Let p be a given prime number greater than 3. Suppose we want to build an elliptic curve of order m, where m satisfies (1) We will use the theory of ECPP to achieve this. The algorithm runs as follows: procedure BuildCurveGivenM(p) 1. compute t = p 1 Gamma ....

J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagstaff, Jr. Factorizations of b n \Sigma 1; b = 2; 3; 5; 6; 7; 10; 11; 12 up to high powers, 2 ed. No. 22 in Contemporary Mathematics. AMS, 1988.

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J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagsta, Jr., Factorizations of b  1, b = 2; 3; 5; 6; 7; 10; 11; 12 up to high powers, 2nd ed., Amer. Math. Soc., Providence, RI, 1988.

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J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagstaff, Jr., Factorizations of b \Sigma 1, b = 2; 3; 5; 6; 7; 10; 11; 12 up to high powers, 2nd ed., Amer. Math. Soc., Providence, RI,

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J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagsta#, Jr., Fac t orizations of b 1, b =2,3,5,6,7,10, 11, 12 up to high powers, 2nd ed., Amer. Math. Soc., Providence, RI,

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J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagsta#, Jr., Factorizations of b b = 2, 3, 5, 6, 7, 10, 11, 12 up to High Powers, third edition, Amer. Math. Soc., Providence, RI, 2002. http://www.ams.org/online bks/conm22/

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J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagsta, Jr., Factorizations of b 1, b = 2; 3; 5; 6; 7; 10; 11; 12 up to High Powers, third edition, Amer. Math. Soc., Providence, RI, 2002. http://www.ams.org/online bks/conm22/

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J. Brillhart, D.H. Lehmer, J.L. Selfridge, B. Tuckerman and S.S. Wagstaff, Factorizations of b \Sigma 1, b = 2; 3; 5; 6; 7; 10; 11; 12 Up to High Powers, Contemporary Mathematics, 22, 2nd ed., AMS, 1988.

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Brillhart (J.), Lehmer (D. H.), Selfridge (J. L.), Tuckerman (B.) et Wagstaff, Jr. (S. S.). -- Factorizations of b n \Sigma 1; b = 2; 3; 5; 6; 7; 10; 11; 12 up to high powers. -- AMS, 1988, 2 'edition, Contemporary Mathematics.

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J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, S. S. Wagstaff, Jr. Factorizations of b n \Sigma 1; b = 2; 3; 5; 6; 7; 10; 11; 12 up to high powers. Contemporary Mathematics, 22, AMS, 1983.

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