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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?
SW VandeCarr said: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!CRGreathouse said:There are more than 1012978181 primes
CRGreathouse said: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 1012978181 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 × 1012978181 and 1.05901756822453 × 1012978181 primes in that range, thanks to Pierre Dusart's tight bounds on pi(x).
SW VandeCarr said: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?
CRGreathouse said:No, it would be just as hard. There's no good way to use that information.
SW VandeCarr said:I appreciate your patience with me. Just one more (two part) related question:
SW VandeCarr said: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?
SW VandeCarr said: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).
SW VandeCarr said: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?
CRGreathouse said: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.
SW VandeCarr said:Thanks for your answers CRGreathouse. That's what I'll do.
CRGreathouse said: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.CRGreathouse said: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.
ramsey2879 said: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.
Count Iblis said: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?
Unknown primes less than the largest known prime refer to prime numbers that are smaller than the largest known prime number, but have not yet been discovered or proven to exist.
To search for unknown primes less than the largest known prime, scientists use advanced algorithms and computer programs designed specifically for finding and verifying prime numbers.
As of now, there is no known limit to the largest prime number. Scientists continue to find larger and larger primes, with the current largest known prime having over 24 million digits.
Finding unknown primes is important for several reasons. It helps us better understand the patterns and properties of prime numbers, and can also have practical applications in fields such as cryptography and computer science.
Yes, there are several ongoing efforts by mathematicians and computer scientists to find unknown primes less than the largest known prime. These efforts use advanced techniques and technology to search for and verify the existence of these elusive numbers.