TITLE: How far is the list of known primes known to be complete?
QUESTION [56 upvotes]: So there is always the search for the next "biggest known prime number". The last result that came out of GIMPS was $2^{74\,207\,281} - 1$, with over twenty million digits. Wikipedia also lists the twenty highest known prime numbers, only the four smallest on that list have fewer than three million digits.
For some while now, I have been wondering about the smaller prime numbers we haven't found. How far up is the list of known primes known to be complete? Since $500$ to $1000$ digit primes are considered safe for the RSA algorithm, I'd assume that it's well below that. How far along the number line have we checked that there are no more primes to be found? How fast is this boundary moving forward, currently? Have we, for instance, checked the primality of all numbers below $10^{100}$, or are we stuck somewhere south of $10^{20}$?

REPLY [7 votes]: Listing primes in order is a fairly trivial problem — in fact, I believe we have programs that can compute lists of primes faster than they can be written to disk, let alone displayed in any human readable format.
The thing is, there are a lot of primes. The entire internet put together probably couldn't store the list of all 20 digit primes.
But that's okay, because we don't need lists of primes — whenever we need primes, we can just generate them.