counting primes

Phil Gibbs

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Apr 1, 1994, 8:12:03 AM4/1/94

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the 1st prime is 2
the 10th prime is 29
the 100th prime is 541
the 1000th prime is 7919
the 10000th prime is 104729
the 100000th prime is 1299709
the 1000000th prime is 15485863

How far can this be extended by computation?

Steve Smith

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Apr 1, 1994, 12:51:00 PM4/1/94

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gi...@eurocontrol.fr (Phil Gibbs) writes:

To the googolplex-th (10^10^100) prime, which if written with
ten-point numerals, could not be contained within the known universe.
This is the primary importance of this famous number.

spu...@pomona.claremont.edu

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Apr 1, 1994, 1:14:26 PM4/1/94

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In article <1994Apr1.1...@fozzie.eurocontrol.fr>, gi...@eurocontrol.fr

(Phil Gibbs) writes:
> the 1st prime is 2
> the 10th prime is 29
> the 100th prime is 541
> the 1000th prime is 7919
> the 10000th prime is 104729
> the 100000th prime is 1299709
> the 1000000th prime is 15485863

There is a function by Gauss (I think) that will approximate the # of
primes below a given n to a ridiculous accuracy like 2 in 10 trillion. Your
function is simply the inverse of that and...
Obpuzzle: Which function, Gauss' or its inverse, will be more accurate?

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David DeLaney

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Apr 1, 1994, 6:55:06 PM4/1/94

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As far as you care to keep computing primes. Eratosthenes' Sieve will do nicely.

Dave "I don't understand your question" DeLaney
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\/ David DeLaney: d...@utkux.utcc.utk.edu; "A penny, a penny, two pence, a penny
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Clifton T. Sharp

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Apr 2, 1994, 12:36:04 PM4/2/94

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Call me a dense set, but I can't see the pattern.

Victor Miller

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Apr 11, 1994, 5:49:58 PM4/11/94

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>>>>> "Phil" == Phil Gibbs <gi...@eurocontrol.fr> writes:
In article <1994Apr1.1...@fozzie.eurocontrol.fr> gi...@eurocontrol.fr (Phil Gibbs) writes:

Phil> the 1st prime is 2
Phil> the 10th prime is 29
Phil> the 100th prime is 541
Phil> the 1000th prime is 7919
Phil> the 10000th prime is 104729
Phil> the 100000th prime is 1299709
Phil> the 1000000th prime is 15485863

Phil> How far can this be extended by computation?

Phil, How far would you like?

the 10000000th prime is 179424671
the 100000000th prime is 2038074743
the 1000000000th prime is 22801763489
the 10000000000th prime is 252097800611
the 100000000000th prime is 2760727302517

[ See Lagarias, Miller and Odlyzko, "Calculating pi(x): the
Meissel-Lehmer method" in Math. Comp. 1985.

--
Victor S. Miller | " ... Meanwhile, those of us who can compute
vic...@ccr-p.ida.org | can hardly be expected to keep writing papers
CCR, Princeton, NJ 08540 | saying 'I can do the following useless
| calculation in 2 seconds', and indeed what
| editor would publish them?" -- Oliver Atkin

Philip Gibbs

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Apr 12, 1994, 5:59:56 AM4/12/94

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In article <VICTOR.94A...@harder.ida.org>, vic...@harder.ida.org (Victor Miller) writes:
> >>>>> "Phil" == Phil Gibbs <gi...@eurocontrol.fr> writes:
> In article <1994Apr1.1...@fozzie.eurocontrol.fr> gi...@eurocontrol.fr (Phil Gibbs) writes:
>
>
> Phil> the 1st prime is 2
> Phil> the 10th prime is 29
> Phil> the 100th prime is 541
> Phil> the 1000th prime is 7919
> Phil> the 10000th prime is 104729
> Phil> the 100000th prime is 1299709
> Phil> the 1000000th prime is 15485863
>
> Phil> How far can this be extended by computation?
>
> Phil, How far would you like?
>
> the 10000000th prime is 179424671
> the 100000000th prime is 2038074743
> the 1000000000th prime is 22801763489
> the 10000000000th prime is 252097800611
> the 100000000000th prime is 2760727302517
>
> [ See Lagarias, Miller and Odlyzko, "Calculating pi(x): the
> Meissel-Lehmer method" in Math. Comp. 1985.
>

That's impressive but I have a quibble. 179424671 is the 9999999th
prime not the 10000000th. The 10000000th is 179424673 according to
UNIX. I used the command "primes 1 179424673 | wc" to check this.
I do agree that 2038074743 is the 100000000th prime but I am
unable to check the others because UNIX says "Ouch." for larger
numbers.

Anyway your response just begs the question: What is
the 1000000000000th prime?

NOTE: followups to sci.math only

Victor Miller

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Apr 15, 1994, 1:04:42 PM4/15/94

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>>>>> "Philip" == Philip Gibbs <ph...@eurocontrol.fr> writes:
In article <1994Apr12....@fozzie.eurocontrol.fr> ph...@eurocontrol.fr (Philip Gibbs) writes:

Philip> In article <VICTOR.94A...@harder.ida.org>,

Philip> vic...@harder.ida.org (Victor Miller) writes:
> >>>>> "Phil" == Phil Gibbs <gi...@eurocontrol.fr> writes:

Philip> That's impressive but I have a quibble. 179424671 is the
Philip> 9999999th prime not the 10000000th. The 10000000th is
Philip> 179424673 according to UNIX. I used the command "primes 1
Philip> 179424673 | wc" to check this. I do agree that 2038074743 is
Philip> the 100000000th prime but I am unable to check the others
Philip> because UNIX says "Ouch." for larger numbers.

Philip> Anyway your response just begs the question: What is the
Philip> 1000000000000th prime?

Philip> NOTE: followups to sci.math only

Phil, I found that my program had an occasional off by one bug.
Here's the corrected list:

10000000 179424673
100000000 2038074743
1000000000 22801763489
10000000000 252097800623
100000000000 2760727302517
1000000000000 29996224275833