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SW VandeCarr
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The largest known primes for some time have all been Mersenne primes. Can it be shown either that there may exist or cannot exist unknown primes less than the largest known prime?
The largest known primes for some time have all been Mersenne primes. Can it be shown either that there may exist or cannot exist unknown primes less than the largest known prime?
That you can even calculate such a number is truly amazing!There are more than 10^{12978181} primes
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).
If I wanted to enumerate the primes over some tractable interval less than the largest known prime , would it be any easier given that at least one prime greater than the upper bound of the interval is known?
No, it would be just as hard. There's no good way to use that information.
I appreciate your patience with me. Just one more (two part) related question:
If I wanted to construct an unbroken sequence of all primes up to some prime p', would this limit be determined only by the limits of computability (brute force and/or algorithmic)? Is there any information regarding about what this limit might be?
1)I'm trying to confirm that all primes up the largest known prime (or any arbitrarily large prime (p')) are in principle computable (Turing definition).
2) Practically there is always a lot of interest in new largest prime, but none (as far as I can tell) as to the longest unbroken sequence of primes from 2 to p' . It seems such lists this would have utility for codes, "random" sequences and simply examining the statistical properties of such sequences. For example, do we know the first 10^6 or 10^7 primes?
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.
Thanks for your answers CRGreathouse. That's what I'll do.
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.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.
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.
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?