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Factorization of the Tenth and Eleventh Fermat Numbers - Brent (1996) (2 citations) Self-citation (Brent) (Correct)
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R. P. Brent, Large factors found by ECM, available by ftp from nimbus.anu.edu.au:/pub/ Brent/champs.ecm .
Factorizations of a^n±1, 13 ≤ a < 100: Update 2 - Brent, Montgomery, Riele Self-citation (Brent) (Correct)
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R. P. Brent, Large factors found by ECM, Computer Sciences Laboratory, Australian National Univ., Canberra, December 1995.
Factorization of the Tenth Fermat Number - Brent (1999) (1 citation) Self-citation (Brent) (Correct)
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R. P. Brent, Large factors found by ECM, Computer Sciences Laboratory, Australian National University, Dec. 1995 (and more recent updates). ftp://nimbus.anu.edu.au/pub/ Brent/champs.ecm .
Some Parallel Algorithms for Integer Factorisation - Brent (1999) (5 citations) Self-citation (Brent) (Correct)
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R. P. Brent, Large factors found by ECM, Oxford University Computing Laboratory, May 1999. ftp://ftp.comlab.ox.ac.uk/pub/Documents/techpapers/ Richard.Brent/champs.txt .
Some Parallel Algorithms for Integer Factorisation - Brent (1999) (5 citations) Self-citation (Brent) (Correct)
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R.P.Brent,Large factors found by ECM, Oxford University Computing Laboratory, May 1999. ftp://ftp.comlab.ox.ac.uk/pub/Documents/techpapers/ Richard.Brent/champs.txt .
Factorizations of Cunningham numbers with bases 13 to.. - Brent, Montgomery, Riele (2001) (1 citation) Self-citation (Brent) (Correct)
....SNFS are large because these methods are only used after ECM has been tried. In fact, since Update 2, MPQS and SNFS did not find any factor with less than thirty digits, because such factors had already been found by ECM. The largest factor found by ECM was a 52 digit factor of 96 98 1 (see [5]) Table 3: Factors Found by Different Methods Pollard Pollard ECM MPQS NFS p Gamma 1 p 1 (30D ) 30D ) Update 1 38 16 69 157 37 Update 2 0 3 151 155 136 Update 3 0 3 423 129 279 5. First Holes A first hole is the first composite number occurring in a table. Thus, each table of ....
R. P. Brent, Large factors found by ECM, Computer Sciences Laboratory, Australian National Univ., Canberra, December 1995. Revision available from ftp:// ftp.comlab.ox.ac.uk/pub/Documents/techpapers/Richard.Brent/champs.txt
Factorizations Of Cunningham Numbers With Bases 13 To.. - Brent, Montgomery, Riele (2000) (1 citation) Self-citation (Brent) (Correct)
....and SNFS are large because these methods are only used after ECM has been tried. In fact, since Update 2, MPQS and SNFS did not nd any factor with less than thirty digits, because such factors had already been found by ECM. The largest factor found by ECM was a 52 digit factor of 96 98 1 (see [5]) Table 3: Factors Found by Di erent Methods Pollard Pollard ECM MPQS NFS p 1 p 1 (30D ) 30D ) Update 1 38 16 69 157 37 Update 2 0 3 151 155 136 Update 3 0 3 423 129 279 5 First Holes A rst hole is the rst composite number occurring in a table. Thus, each table of factorizations is ....
R. P. Brent, Large factors found by ECM, Computer Sciences Laboratory, Australian National Univ., Canberra, December 1995. Revision available from ftp:// ftp.comlab.ox.ac.uk/pub/Documents/techpapers/Richard.Brent/champs.txt
Recent Progress and Prospects for Integer Factorisation Algorithms - Brent (2000) (4 citations) Self-citation (Brent) (Correct)
....too seriously. 2. The largest factor known to have been found by ECM is the 54 digit factor 484061254276878368125726870789180231995964870094916937 of (6 43 1) 42 1, found by Nik Lygeros and Michel Mizony with Paul Zimmermann s GMP ECM program [63] in December 1999 (for more details see [9]) 3.2 Parallel Distributed Implementation of ECM ECM consists of a number of independent pseudo random trials, each of which can be performed on a separate processor. So long as the expected number of trials is much larger than the number P of processors available, linear speedup is possible by ....
.... 50 52 54 D Figure 1: Size of factor found by ECM versus year Recent Progress and Prospects for Integer Factorisation Algorithms 7 Figure 1 shows the size D (in decimal digits) of the largest factor found by ECM against the year it was done, from 1991 (40D) to 1999 (54D) historical data from [9]) 3.4 Extrapolation of ECM Records 1990 1992 1994 1996 1998 2000 2002 6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 7.6 # D Figure 2: # D versus year Y for ECM Let D be the number of decimal digits in the largest factor found by ECM up to a given date. From the theoretical time bound for ECM, ....
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R. P. Brent, Large factors found by ECM, Oxford University Computing Laboratory, March 2000. ftp://ftp.comlab.ox.ac.uk/pub/Documents/techpapers/ Richard.Brent/champs.txt .
Recent Progress and Prospects for Integer Factorisation Algorithms - Brent (4 citations) Self-citation (Brent) (Correct)
....too seriously. 2. The largest factor known to have been found by ECM is the 54 digit factor 484061254276878368125726870789180231995964870094916937 of (6 43 1) 42 1, found by Nik Lygeros and Michel Mizony with Paul Zimmermann s GMP ECM program [63] in December 1999 (for more details see [9]) 3.2 Parallel distributed implementation of ECM ECM consists of a number of independent pseudo random trials, each of which can be performed on a separate processor. So long as the expected number of trials is much larger than the number P of processors available, linear speedup is possible by ....
....the expected run time on a MIMD parallel machine with P processors is T P = T 1 =P O(T 1=2 1 ) 1) 3. 3 ECM factoring records Figure 1 shows the size D (in decimal digits) of the largest factor found by ECM against the year it was done, from 1991 (40D) to 1999 (54D) historical data from [9]) 3 1990 1992 1994 1996 1998 2000 2002 38 40 42 44 46 48 50 52 54 D Figure 1: Size of factor found by ECM versus year 3.4 Extrapolation of ECM records Let D be the number of decimal digits in the largest factor found by ECM up to a given date. From the theoretical time bound for ....
R. P. Brent, Large factors found by ECM, Oxford University Computing Laboratory, March 2000. ftp://ftp.comlab.ox.ac.uk/pub/Documents/techpapers/Richard.Brent/ champs.txt .
Factorization of the Tenth Fermat Number - Brent (1998) (1 citation) Self-citation (Brent) (Correct)
....factorization of F 11 was announced in [9] F 11 = 319489 Delta 974849 Delta 167988556341760475137 Delta 3560841906445833920513 Delta p 18 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 B 1 = 370; 000, D = 255, e = 6, we found the 40 digit factor p 0 40 = 9409853205696664168149671432955079744397 of p 252 Gamma 1, where p 252 is the largest prime factor of F 10 . See ....
R. P. Brent, Large factors found by ECM, Computer Sciences Laboratory, Australian National University, Dec. 1995 (and more recent updates). ftp://nimbus.anu.edu.au/pub/ Brent/champs.ecm .
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