At what point are there gaps in the sequence of known prime numbers.

Question

I'm not sure how to properly word this question so please bear with me. If you look at the beginning of the prime number list it's obvious that the sequence is complete. At least up to a certain point. I was reading an article about a recently discovered prime number and it stated that Mersenne Prime's are currently being targeted. They referred to them as low hanging fruit and stated that many numbers are not being checked. So my question is at what point do we start having gaps in the "sequence" of prime numbers.

Answer 1

It's not really a thing that's being kept track of. In general, we don't really store large primes; rather, we just recompute them when we need to. This is because it's cumbersome to store all the primes we know--and we know a lot of them. For instance, there are $37,607,912,018$ primes less than $1 \times 10^{12}$.

For pretty much any practical activity that requires use of large primes, it's easier to test for primality or simply generate primes than to store them and look them up. In this sense, the largest value of $x$ for which we know every single prime less than $x$ is really a function of who has the largest list on their hard drive.

And we know that it's a just a matter of generation (as opposed to a matter of gaps (ha!) in knowledge) because if we did know the largest $x$ for which we know every single prime less than $x$, then it would be trivial to just test primes above $x$ using sieve methods, and then we'd have a new $x$.