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/ann2_673M_1.html
M673 at 203 digits had two known prime factors, 581163767 and 41283139633378645724930694480520226273492263, a 44digit factor discovered by ECM using mprime. The remaining 151digit composite cofactor split into a 59digit prime factor and a 92digit prime factor listed on the webpage given above. Using prime95 (mprime) ECM, I calculate that the 59digit 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 55digits. 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.
