On March 27, 2010, Michael Vang found a 54-digit factor of F12 using GMP-ECM. F12 is the 12th Fermat number 2^(2^12)+1 = 2^4096 + 1. Fermat numbers are of special interest for integer factorization, and F12 is the smallest unfactored Fermat number. F11 = 2^2048+1 was completely factored in 1988 by Brent and Morain, and F10 = 2^1024+1 was completely factored in 1995 by Brent [1]. About F12. Before Michael Vang's discovery, we knew the following factors: F12 = 114689 * 26017793 * 63766529 * 190274191361 * 1256132134125569 * c1187 where c1187 denotes a composite number of 1187 digits. Michael Vang found a 54-digit factor of c1187 [3]: c1187 = 568630647535356955169033410940867804839360742060818433 * c1133 where c1133 is still a composite number, of 1133 digits. This 1133-digit cofactor is still out of reach of factorization algorithms such as NFS (which was used recently to factor the 232-digit number RSA-768), thus the best hope to complete the factorization of F12 is that another factor is found by ECM. About GMP-ECM. GMP-ECM is a program for integer factorization using the Elliptic Curve Method (ECM). ECM was invented by Hendrik W. Lenstra, Jr. The first version of GMP-ECM was written by Paul Zimmermann, using the GNU MP (GMP) library for efficient arithmetic. Several people contributed to GMP-ECM, in particular Pierrick Gaudry developed some efficient assembly code for Stage 1, and Alexander Kruppa developed (among other contributions) some special-purpose code to factor Fermat numbers. This part of the ECM code was used by Michael Vang. For more on GMP-ECM or to download the program, visit [2]. About the lucky curve. The curve used by Michael Vang has sigma=1428526317, and the corresponding group order is: 2^4 * 3^2 * 7 * 17 * 293 * 349 * 8821 * 23753 * 65123 * 2413097 * 9027881 * 23759413 * 45947380867 The second-largest group order factor 23759413 is below the stage 1 bound B1=43000000 used by Michael Vang, and the largest factor 45947380867 is below the stage 2 bound B2=199103726650 used by GMP-ECM for this value of B1. [1] http://prothsearch.net/fermat.html [2] http://www.loria.fr/~zimmerma/records/ecmnet.html [3] http://www.mersenneforum.org/showthread.php?p=209724