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Old 2003-03-17, 23:24   #1
philmoore
 
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"Phil"
Sep 2002
Tracktown, U.S.A.

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Default M673 completely factored

It was announced today that M673, the smallest Mersenne number which had not been completely factored into its prime factors, was factored by the Number Field Sieve NFSNET group:

http://www.nfsnet.org/ann-2_673M_1.html

M673 at 203 digits had two known prime factors, 581163767 and 41283139633378645724930694480520226273492263, a 44-digit factor discovered by ECM using mprime. The remaining 151-digit composite cofactor split into a 59-digit prime factor and a 92-digit prime factor listed on the web-page given above. Using prime95 (mprime) ECM, I calculate that the 59-digit factor would have turned up approximately once every 700,000 curves with the stage 1 bound set to 44,000,000. Not real encouraging, but then with the 5596 curves currently listed on the status page, that still translates into a 1 in 125 chance of finding a huge factor by ECM standards: the current record is 55-digits. Hopefully, ECM enhancements can double the efficiency of the program.

M713 is now the smallest Mersenne number which is not completely factored. The exponent 713=23*31 is composite, and M739 is the smallest with a prime exponent.
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