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MathGroup Archive 2007

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Re: Re: PrimePi and limit of argument

  • To: mathgroup at smc.vnet.net
  • Subject: [mg78128] Re: [mg78087] Re: [mg77911] PrimePi and limit of argument
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Sat, 23 Jun 2007 07:15:53 -0400 (EDT)
  • References: <7566193.1182423757619.JavaMail.root@m35> <200706221049.GAA16611@smc.vnet.net>

That's not quite right: in the case of solving polynomial equations  
in terms of radicals there is a mathematical limitation, in the other  
case there is a limitation of the particualr implementation. The  
value of PrimePi[10^15]  is well known, in fact it is:

29844570422669

In Mathemaitca you can get a pretty good approximation by evaluating:


Round[LogIntegral[10^15]]
29844571475288


Andrzej Kozlowski


On 22 Jun 2007, at 19:49, DrMajorBob wrote:

> If you decide to compute PrimePi[100] by hand, you might take a  
> piece of
> paper and write down the primes up to 97, then count them. If you  
> try the
> same method for PrimePi[10^15], you'll need a bigger piece of paper.
>
> But you'll need a lot MORE than a bigger piece of paper -- you'll  
> need a
> smarter algorithm, or you'll never live long enough. And that's what
> Mathematica is telling you; the PrimePi method has an upper ceiling,
> independent of how big your machine might be.
>
> You may as well demand a general solution in radicals for 7th-degree
> polynomials.
>
> 7 isn't a large number, but even so, it can't be done... even if your
> machine is bigger than the universe.
>
> Bobby
>
> On Tue, 19 Jun 2007 05:47:55 -0500, Robert Pigeon
> <robert.pigeon at videotron.ca> wrote:
>
>> Hello all,
>>
>>   I was playing around with the function PrimePi[] and trying  
>> different
>> arguments. When I tried PrimePi[10^15] I got the error message saying
>> that
>> the argument is too large for this implementation. I know that it  
>> is a
>> large
>> number.! When I use 10^14 as the argument I get an answer, it takes a
>> while
>> but I get an answer.
>>
>>  I tried this on a PC running Vista Home Premium 64-bit with  
>> Mathematica
>> 6.
>> Then I tried the same thing under Windows XP 32-bit. There was no
>> difference, I got an answer for 10^14 and same error message with  
>> 10^15.
>>
>>
>> My question is: I thought that with a 64-bit computer I could use  
>> larger
>> numbers.! Maybe I am misunderstanding something here, so please  
>> help me
>> understand J
>>
>>
>> Thanks,
>>
>>
>> Robert
>>
>>
>> Robert Pigeon
>>
>>
>
>
>
> -- 
> DrMajorBob at bigfoot.com
>



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