Palaiseau, 030716 this is a follow up to my 030606 email on the fast version of ECPP. In the meantime, I have proven the primality of two larger numbers, namely x^y+y^x for (x, y)=(2589, 218) -- 6055 decimal digits -- and (x, y)=(2551, 622) -- 7127 decimal digits, taken from Paul Leyland's table: http://research.microsoft.com/~pleyland/primes/xyyx.htm The CPU time for that number was roughly 625 GHz-days (on 6 biprocessors Xeon), as opposed to the preceding (monoprocessor) record of Hans Rosenthal, a 6959dd number using 738 GHz-days. A very preliminary version of the paper is available from my web page: http://www.lix.polytechnique.fr/Labo/Francois.Morain F. Morain