
New record prime: 2^{74,207,281}1 with 22,338,618 digits by Cooper, Woltman, Kurowski, Blosser & GIMPS (7 Jan 2016). 

It is often wondered what is the longest list of consecutive primes, starting at two, that has ever been found? Sometimes it is asked in a different manner: "what is the smallest number n such that it is not known whether or not n is prime?" (Of course there are infinitely many primes, so there is no theoretical limit to the length of such a list.) Perhaps the longest lists ever calculated (but not all stored) are those corresponding to the mximal prime gape (and twin prime constant) projects. See Nicely's lists. At the time I last updated this page, these projects had found (but not stored) all the prime up to 10^{18}, but not yet to 10^{19}. The problem with answering this question is small primes are too easy to find. The can be found far faster than they can be read from a hard disk, so no one bothers to keep long lists (say past 10^{9}). Long lists just waste storage, and if placed on the Internet, they just waste bandwidth. Nevertheless, due to popular demand, I have placed several lists on this site, such as the first 100,008 primes and the first fifty million primes. If you want an even longer list, run a sieve program on your machine. Folks quite regularly resieve to find all the primes up to 1,000,000,000,000, this should take well less than a minute. The answer to the second form of the question is similar. If we could
give the smallest number n such that it is not known whether or
not n is prime, then someone could check the next million primes
in about a second of computer time (at most!). 
Another prime page by Chris K. Caldwell <caldwell@utm.edu> 