Joe & Mike w/Glacier, 01/25/2009
December 6th, 2009: Using WinPFGW, GMP-ECM, and GGNFS we found the following new record for nine consecutive factorizations at 804 digits. This is a significant step above the prior records. See Largest Consecutive Factorizations for more information.
N = (y-232) (y-242) / 55440
- y = (t(5t+9)/2 - 31)2
- t = 5x3 - x2 - x - 1
- x = 2310 (1030+40790547)+5
- N-k is factored for k=0..8.
It is algebraically smooth for N, N-3, N-4, and the substitution 't' is so we have only five composites with SNFS ~200 difficulty within those algebraic factorizations.
WinPFGW was used to find x candidates with easy N-k factorizations for several k, then GMP-ECM was used to further inspect those candidates until a complete k=0..8 set was found (i.e. all algebraic factors with degree higher than 6 completely factored). Most of the composites were resolved using ECM, then SNFS was used to finalize the remaining C169=p57*p112 and C199=p58*p142.
Thanks to Jens Kruse Anderson for maintaining the record pages and to David Broadhurst for the friendly competition and sharing of ideas.
Some links I recommend if you are interested in number theory:
