Currently, what is the largest publicly known prime number such that all prime numbers less than it are known?

So recently, an absurdly large prime number was found, but a lot of prime numbers less than it are still not known. I am wondering up to where we know all the primes.

I put "currently publicly known" because there is a chance that some government agency has a longer list for crypto reasons or something like that.

Solution 1:

The most efficient known way (please correct me if I'm wrong) to generate a list of consecutive primes from $2$ to $n$ is the Sieve of Eratosthenes, which in an optimized implementation (at least relying on what's written in Wikipedia) requires $O(n)$ time and something like $O(n^{1/2+\epsilon})$ memory. Given current computing abilities, I'd guess your prime is somewhere between $2^{50}$ and $2^{60}$.

Edit to clarify: Asking for an exact answer is meaningless, because given a prime of that size, it's pretty fast to calculate the next one.

Edit 2 to answer your question with another question. What do you mean by "known"? Do you want them all to be written down in a physical list? By the prime number theorem, there are about $\frac n {\log n}$ primes up to $n$, so you'd need a pretty big piece of paper (or hard drive) to write down the primes up to $2^{60}$ :)

Solution 2:

I don't think you can pinpoint such a prime. If you had a candidate, it wouldn't be too hard to determine the next larger prime. There are just too many of them.