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[SUP]82,589,933[/SUP]-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?

[QUOTE=lukerichards;513409]We know all primes, p such that p < 10: 2, 3, 5 7

We do not know all primes, r, such that r < 2[SUP]82,589,933[/SUP]-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?[/QUOTE]
If memory serves, the [i]number[/i] of primes less than 10[sup]27[/sup] is known, though I doubt they are all listed.

Pari-GP offers values of [b]primelimit[/b], the upper bound on the list of precomputed primes, of 2[sup]32[/sup] or 2[sup]64[/sup], according to whether you have a 32-bit or 64-bit machine.

I'm guessing the precomputation of all primes up to 2[sup]64[/sup] might take a while, and the resulting vector of primes might use up a bit of RAM...

[QUOTE=lukerichards;513409]We know all primes, p such that p < 10: 2, 3, 5 7

We do not know all primes, r, such that r < 2[SUP]82,589,933[/SUP]-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?[/QUOTE]

That is an interesting question. I believe it is a number "slightly" over 4*10[SUP]18[/SUP]

Tomas Oliveira e Silva computed all prime numbers up to that limit exactly 7 years ago in April 2012. See [URL="http://sweet.ua.pt/tos/gaps.html"]http://sweet.ua.pt/tos/gaps.html[/URL]

Perhaps –because it is relatively easy to do– someone else with the right equipment has computed 10[SUP]15[/SUP] over that limit.

[QUOTE=lukerichards;513409]What estimates can you make for the size of q?[/QUOTE]
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?

Remeber that the universe has *only * [B]10[/B]^78 to [B]10[/B]^82 atoms.

[QUOTE=uau;513509]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?[/QUOTE]

It wasn't supposed to be a meaningful question, that's why it was posted to "Fun Stuff 》Puzzles".

[QUOTE=firejuggler;513510]Remeber that the universe has *only * [B]10[/B]^78 to [B]10[/B]^82 atoms.[/QUOTE]

Struggling to identify the significance of this point, re the original question.

[QUOTE=firejuggler;513510]Remeber that the universe has *only * [B]10[/B]^78 to [B]10[/B]^82 atoms.[/QUOTE]

[QUOTE=lukerichards;513534]Struggling to identify the significance of this point, re the original question.[/QUOTE]

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.

[QUOTE=CRGreathouse;513537]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.[/QUOTE]

Now that is a fascinating way to look at it.

[QUOTE=CRGreathouse;513537]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.[/QUOTE]

That is a really high number but the practical limit is way less than that.

I don't have any special predictive ability but I would say that the actual limit which is 4*10[SUP]18[/SUP] (which was reached on April 2012) can be a[FONT="Arial Black"][U]t most[/U][/FONT] be squared to 1.6*10[SUP]37[/SUP] and I don't expect anyone who is alive now will be alive if and when this happens.

[QUOTE=rudy235;513413]That is an interesting question. I believe it is a number "slightly" over 4*10[SUP]18[/SUP]

Tomas Oliveira e Silva computed all prime numbers up to that limit exactly 7 years ago in April 2012. See [URL="http://sweet.ua.pt/tos/gaps.html"]http://sweet.ua.pt/tos/gaps.html[/URL]

Perhaps –because it is relatively easy to do– someone else with the right equipment has computed 10[SUP]15[/SUP] over that limit.[/QUOTE]Curiously, this is still less than 2^64, the maximum value for [b]primelimit[/b] offered by Pari-GP for 64-bit machines.

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