20190411, 13:14  #1 
"Luke Richards"
Jan 2018
Birmingham, UK
2^{5}×3^{2} Posts 
Known primes
We know all primes, p such that p < 10: 2, 3, 5 7
We do not know all primes, r, such that r < 2^{82,589,933}1 Therefore there is a prime, q, such that all primes less than q are known and proven, but there are primes greater than q not known. What estimates can you make for the size of q? 
20190411, 13:51  #2  
Feb 2017
Nowhere
2·2,081 Posts 
Quote:
PariGP offers values of primelimit, the upper bound on the list of precomputed primes, of 2^{32} or 2^{64}, according to whether you have a 32bit or 64bit machine. I'm guessing the precomputation of all primes up to 2^{64} might take a while, and the resulting vector of primes might use up a bit of RAM... 

20190411, 13:56  #3  
Jun 2015
Vallejo, CA/.
2·5·97 Posts 
Quote:
Tomas Oliveira e Silva computed all prime numbers up to that limit exactly 7 years ago in April 2012. See http://sweet.ua.pt/tos/gaps.html Perhaps –because it is relatively easy to do– someone else with the right equipment has computed 10^{15} over that limit. Last fiddled with by rudy235 on 20190411 at 14:07 Reason: spelling 

20190412, 14:31  #4 
Jan 2017
2·43 Posts 
I'm not sure this is a meaningful question. People don't actually use such lists of primes. What does it mean for a prime to be "known"? That someone has in principle run a probabilistic primality test on it, then thrown away the result?
For comparison, what do you think is the largest number such that someone has counted up to it, but no one has counted further? 
20190412, 15:05  #5 
Apr 2010
Over the rainbow
4724_{8} Posts 
Remeber that the universe has *only * 10^78 to 10^82 atoms.
Last fiddled with by firejuggler on 20190412 at 15:06 
20190412, 18:21  #6  
"Luke Richards"
Jan 2018
Birmingham, UK
100100000_{2} Posts 
Quote:


20190412, 18:22  #7 
"Luke Richards"
Jan 2018
Birmingham, UK
2^{5}·3^{2} Posts 

20190412, 18:58  #8 
Aug 2006
3×1,987 Posts 
It gives a hard upper bound on the number of "known" primes, depending on the representation used of course. Say an atom represents a bit and the primes are written in binary, then you can't know all the primes higher than ~ 7e81 because you run out of space.

20190412, 19:21  #9  
"Luke Richards"
Jan 2018
Birmingham, UK
2^{5}·3^{2} Posts 
Quote:


20190412, 20:22  #10  
Jun 2015
Vallejo, CA/.
2×5×97 Posts 
Quote:
I don't have any special predictive ability but I would say that the actual limit which is 4*10^{18} (which was reached on April 2012) can be at most be squared to 1.6*10^{37} and I don't expect anyone who is alive now will be alive if and when this happens. 

20190413, 15:02  #11  
Feb 2017
Nowhere
2·2,081 Posts 
Quote:
This leads me to wonder  what is the largest value for primelimit people actually use commonly? (The default is 500k.) 

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