Section 16.3 due Dec. 3rd

(Difficult)
I was a little confused by the example on pg. 357-358. The example used factorials multiplied by P, such as 10!P. In the previous examples, we were simply adding P + P + P... until we found a value m for which mP = infinity. Do we only find mP = infinity for m, or at every multiple of m. If not, are we guaranteed to find a solution if we are working instead with factorials? That is, will the value of m always be an integer equal to some factorial value? Intuitively it seems it must be at every multiple of m for the factorial method to work, but I don't recall reading this anywhere in this lecture or the previous.

(Reflective)
I found it interesting that the factoring method of elliptic curves was so similar to the p-1 method. It never would have occurred to me just from reading about elliptic curves that the factoring methods were so closely related. In the lecture it mentioned though that elliptic curves are most effective on large composite numbers with one relatively small prime factor. Each method we have discussed has its own advantages and disadvantages, but this made me think about the methods of choosing primes. When we talked about primality testing, we read about many probabilistic tests, but only one definitive method of testing for primality. If a number is choosen as a prime p, but is actually a psuedoprime, would this leave it vulnerable to any factoring methods?