What is the longest list of primes? 
(from the Prime Pages' list of frequently asked questions)
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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 1018, but not yet to 1019.

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 109).  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, the largest which lists the first 100,008 primes--as far as my conscience would allow.  There are much longer lists available.

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!).

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Another prime page by Chris K. Caldwell <caldwell@utm.edu>