Originally posted by vjs
Also I'm still not sure about why we have to do all of the 50's before the 55's etc.
First of all: I'm no ECM-expert or such, but I think I can answer this question

You don't have to do all the 50's before the 55's etc., but it is recommended. This is because you would need more curves on a 55-digit-level to find for example a 49-digit-factor than on a 50-digit-level.

At all it's just a question of efficiency.

(Sure, if you know that no factor below 50-digits exists you just can start with a 55-digit-level )

ciao
Zahme