200 digits is pretty much impossible with a standard computer if you observe some basic rules, those rules imposing the general idea of these 2 points:
1. Not too close to the square root so that an algorithm working from the square root out can easily find them
2. Not too far apart so the smaller prime number becomes a weak target.
I had a class in my first year of my degree where we had to crack each others encryption, where we gave out the public key pq with an encoded message and we had to crack it by finding p and q.
Back then anyone with a pq > about 105, I couldn't crack unless their numbers broke 1 of the 2 rules above. Or this 1 case where it was rather easy to crack because the number started with a 1, had about 70 0's following it, followed by about 7 random numbers, followed by about another 70 0's followed by about 14 random numbers or so.
I was using a 2.6 GHz Celeron to crack them from my laptop, these days I have a much faster CPU, but no idea how much of a difference that'd make on my ability to crack them.
I'm so geeky I got really excited about making this signature in SVG until it occurred to me HTML would obviously be turned off ¬_¬'