I am playing around and discovered a possible prime number with more than 3000 digits:
2 ^ 10005 * 210 -19 [3015 digits] is probably prime to a certainty of 99.21875%.
I haven't verified this, but it is the largest possible prime number I personally have found. Of course the world record has 22 million digits so this is nothing.
I will post bigger ones if I find them.
2 ^ 10017 * 210 -17 [3018 digits] is probably prime to a certainty of 87.5%.
This is a huge number that start with 549143 and ends with 525103.
I turned down the certainty factor to speed up the process. It takes about 15 seconds to check each candidate. To verify this to 100% certainty could take hours, and I'm not interested in doing that right now.