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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: [mg78141] Re: [mg78087] Re: [mg77911] PrimePi and limit of argument
  • From: DrMajorBob <drmajorbob at bigfoot.com>
  • Date: Sat, 23 Jun 2007 07:22:33 -0400 (EDT)
  • Organization: Deep Space Corps of Engineers
  • References: <7566193.1182423757619.JavaMail.root@m35> <200706221049.GAA16611@smc.vnet.net> <25396560.1182560007011.JavaMail.root@m35>
  • Reply-to: drmajorbob at bigfoot.com

My point was that increasing the address space -- like getting a bigger  
piece of paper -- doesn't solve every big problem, as if by magic.

"Look! I can calculate 500 factorial! So I can solve equations in that  
many unknowns, too, right? I can find the 500 factorial-th prime, right?"

Well... no. Sorry.

Mathematica will, for the most part, solve the same problems on a 64-bit  
machine as it does on a 32-bit machine. It won't run out of memory as  
often, perhaps, but the algorithms don't suddenly change.

Bobby

On Fri, 22 Jun 2007 19:40:56 -0500, Andrzej Kozlowski <akoz at mimuw.edu.pl>  
wrote:

> *This message was transferred with a trial version of CommuniGate(tm)  
> Pro*
> 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
>>
>
>



-- 
DrMajorBob at bigfoot.com


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