Factorization probability I'm interested in factoring large numbers. I would like to know that if I take a normal laptop and implement for example general or special number sieve and let it run all the time, what would be the probability that I will find a new prime factor for some Fermat number?
 A: If you literally mean "all the time", then the probability is $1$. Prime factorization is a necessarily finite process. You end up with some answer at the end, even if it's that the only prime factor of the number is the number itself.
If you don't literally mean "all the time", but perhaps only "a few hours every day for a year", then the probability directly depends on how fast your laptop searches through possible factors.
A: There is no fixed probability of finding a factor and depends on the number you select and the processing speed of your computer. It also depends on how long you are going to run it. You can see this link for known factors and how much computational power it actually takes o find them.
A: It's highly unlikely. Looking at the page Dietrich linked, you will see that a factorization for $m = 12, k = 7$ discovered by Pervushin and Lucas in 1877 is still "known to be incomplete"! After more than a century! I've been alive almost a century but already three laptops have died on me, and I didn't even subject them to such intense number crunching.
If you're serious about this, you're going to need a good, strong, sturdy desktop computer devoted to running the most efficient algorithm you can get your hands on. And also some kind of back-up system.
Good luck with that.
