Section 3.10

I was wondering since this method is more general does that mean that we might as well jump right into this method when trying to decide if a number is a square or is it worth it to try the method we used last time and then if we have to come here?  

I am also a little confused on parts 4 and 5 of the jacobi symbols part, it seems like it might be building on what we did last time, but I got a little lost reading about it.

In the end it seems like everything we do, is basically saying that its still really hard to factor n.  You have some things that are like factoring n, but we still cant do those things either.  Kind of a frustrating approach to solving a problem, probably worse for those more invested in it than I...