Known primes

[CRGreathouse]

Originally Posted by rudy235

That is an interesting question. I believe it is a number "slightly" over 4*10¹⁸

We've certainly gone over 6787988999657777797 > 6.7*10¹⁸ on these very boards. Someone with more patience can look up the exact bounds.

[retina]

The original question is not clear. The word "known" is undefined in the context.

However, if we assume "known" to mean something like "is stored somewhere (in a brain, on an HDD, or anywhere retrievable)" then I expect there is no way we (as a collective global group) "know" all the primes up to 10¹⁸. The storage requirements are too extreme for any one entity to bother with. Disparate groups and individuals might have particular ranges stored in various places, but AFAICT there has been no coordination to close all the gaps and ensure a contiguous range.

While it might be true that all primes have been computed or identified up to 10¹⁸ (or more perhaps?), I don't see any evidence that those primes have been permanently recorded anywhere. What would people do with such a huge DB anyway? It is probably easier to simply recompute any range anyone is interested in, than trying to read the primes out of some enormous storage array.

[rudy235]

Originally Posted by retina

The original question is not clear. The word "known" is undefined in the context.

However, if we assume "known" to mean something like "is stored somewhere (in a brain, on an HDD, or anywhere retrievable)" then I expect there is no way we (as a collective global group) "know" all the primes up to 10¹⁸. The storage requirements are too extreme for any one entity to bother with. Disparate groups and individuals might have particular ranges stored in various places, but AFAICT there has been no coordination to close all the gaps and ensure a contiguous range.

While it might be true that all primes have been computed or identified up to 10¹⁸ (or more perhaps?), I don't see any evidence that those primes have been permanently recorded anywhere. What would people do with such a huge DB anyway? It is probably easier to simply recompute any range anyone is interested in, than trying to read the primes out of some enormous storage array.

What you say is very true, the primes are probably deleted as soon as the researcher finishes mining the data he/she is looking for.

For instance, an exact count of Twin primes, Cousin, primes, sexy primes, gaps between pairs of Twin primes, exact count of triplets p, p+2 p+6 and p, p+4, p+6, quadruplets, Sophie Germain prime pairs, Cunningham prime pairs, etc.

However, the fact they are NOT stored is a convenience factor. As you well say they can be recomputed and there is very little need to keep several exabytes of frozen storage.

[a1call]

Originally Posted by Dr Sardonicus

Curiously, this is still less than 2^64, the maximum value for primelimit offered by Pari-GP for 64-bit machines.

This leads me to wonder -- what is the largest value for primelimit people actually use commonly? (The default is 500k.)

it seems to me that there actually isn't any way of changing the actual primelimit in Pari-GP.

This default is only used on startup: changing it will not recompute a new table.

https://pari.math.u-bordeaux.fr/dochtml/html-stable/

You can use the default() function to change the primelimit value, but this does not seem to actually do anything other than change the system variable. No new primes are added to the "table". You will have to manually compute and then add the primes to the list.

I'd love to be corrected on this.


Chances are that the available memory on the PC will determine the real upper limit.

Here is a relevant thread:

Largest Primorial that would Fit on a Terabyte Drive.

[a1call]

So I did some experimentation and these seem to be the results.
Increasing the primelimit to 19^6, did not add any primes to the table. Afterwards using a factor function to the limit of 19^6 finished in a few minutes and did not add any calculated:used primes to the table. Then I tried to do a forprime 500000 to 19^6 and add the primes using addprimes function. After a few hours I gave up and tried smaller upper bound of 3*19^5 which probably took about couple of hours to conclude.