Unknown primes less than largest known prime?

I'm not sure you can even prove that there aren't other Mersenne primes between the those found.

 

If you're talking only about other Mersenne primes, mgb_phys answered it: we have no idea. But to answer your literal question:

There are more than 10^12978181 primes* beneath the largest known Mersenne prime. If storing each prime takes 1 bit, there has not been enough storage media produced on Earth through all history (hard drives, DVDs, clay tablets) to hold all these primes.

So yes, there are unknown primes beneath the largest known prime.

* In particukar, there are between 1.05901756822452 × 10^12978181 and 1.05901756822453 × 10^12978181 primes in that range, thanks to Pierre Dusart's tight bounds on pi(x).

 

That you can even calculate such a number is truly amazing!

 

No, it would be just as hard. There's no good way to use that information.

 

I don't mind answering your questions.

I don't understand what you're asking. But practically, if you want to make a huge list of primes, the issue you'll run into is storage: the memory needed for an efficient search (say, at least 20 * sqrt(n)/log(n) bytes to search up to n), and the hard drive (or whatever) space to keep the primes you find (say, 16 * n / log(n) bytes to store the primes up to n, or an eighth that amount for just the differences). So if you wanted a list of primes up to 10^20, you'd need several million terabytes of hard drive space (tens of millions unless you just store the differences).

 

They are.

There are algorithms that, given enough memory, can compute primes up to n faster than iterating through 1, 2, ..., n. So storage becomes the limiting concern there. You could fill a 250 GB hard drive with primes, but what then? It's often faster to generate them and use them on the fly anyway -- don't bother with the (slow) hard drive.

 

May I ask what your application is?

 

Just out of curiosity and possibly to help the originator of this thread, can someone point to a source for such a list generating algorithm. I know of prime lists of the first n primes, n = 10,000 or lower are the typical lists of primes that I have seen, but as to the quickest or most powerful list generator, I have no idea of where to look.

 

The Sieve of Atkin is the asymptotically fastest method I know. Bernstein has an excellent implementation on his site.

 

Isn't it easier to use the formula by Riemann that gives an exact formula for pi(x) as a series involving the zeroes of the Riemann zeta function?

 

Not if you want an exact answer.