During beta testing of GMP-ECM 6.0, a new ECM record prime factor was
found by Bruce Dodson on an IBM/linux workstation (with 2.8Ghz xeon chips,
2Gb memory) at Lehigh University. The new record has 59-digits,
p59 = 20131492120828919814484857298874674155298711142397769181347,
and completes the factorization of the remaining 162-digit composite cofactor
of the Most Wanted number 10^233-1, as p59*p103. The first step limit used was
B1=260M, optimal for finding p60-factors; and the default step 2 limit used
1.2Gb memory. The lucky value of sigma was sigma = 4114600819, which
determines an elliptic curve by the parameterization described by Brent
(and recalled in Dodson's NMBRTHRY post of a previous ECM record with
57-digits, 29 July 2003). A Magma procedure written by Allan Steel
was used to find the order of the elliptic curve, which gave factors
2^3 3 5 29 43 883 73327 76603 (small, < 10^6)
10038697 46168679 59527159 134939023 (medium, below B1)
7285852169 (large, between B1, B2).
This curve would have missed the p59 factor with p55-limits of B1=110M, since the
2nd largest prime factor is 134.9M, but the step 2 factor of the curve order is
not especially large, with 7285M < 30*B1 << B2. An extended run on a beowulf
cluster with 50-limits produced a large number of hard factors during 2004,
including five of the top ten for the year, but none larger than 55-digits.
Perhaps this indicates that breaking the 60-digit barrier is more likely with
the larger limits (p55 or p60) for which the current GMP-ECM 6.0 substantially
improves upon the previous version 5.0.3.