
New record prime (GIMPS): 2^{82,589,933}1 with 24,862,048 digits by P. Laroche, G. Woltman, A. Blosser, et al. (7 Dec 2018). 

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> 