You're quite right but I never actually meant to give a solution to the 200-digit long number, I was only recapping the typical algorithms one can come up with, pretenselessly ;-)...
There must be an option with n! numbers because they grow so quickly - I thought that n!+1 might a prime number but an easy counter-example is n=4 (which'd give n!=24 hence n!+1=25 = 5*5, barely what you can call a prime number...)