Quote:
Originally Posted by jocelynl
take M67  1 (2^672)
all factors of this are: (2,3,3,7,23,67,89,683,20857,599479)
so it contains 67
and it`s the same for all mersenne.

So, are you saying your method of finding factors of Mersenne numbers requires completely factoring (Mersenne number  1) in each case?
That task, in general, is
not faster than LucasLehmer. Sure, it's costeffective to use certain amounts of trial factoring, P1, and ECM to try to find factors of Mersenne numbers before performing the LL. But for any large Mersenne number, those certain points take us only a small fraction of the way through all possible factors. Beyond those points (which differ according to the particular method), it is faster to prove a Mersenne number composite by LL than by finding a factor.