[racket] Math library ready for testing

I clicked the link, and the result shows 78498 which jives with nth-prime

On Tue, Dec 11, 2012 at 8:31 AM, Phil Bewig <pbewig at gmail.com> wrote:

> There must be an error in the prime-counting function. According to <a
> href="
> http://www.wolframalpha.com/input/?i=how+many+primes+less+than+a+million">Wolfram|Alpha</a>,
> there are 79486 primes less than a million, not 78497.
>
> I don't use Racket, but I do have lots of Scheme code that computes with
> prime numbers at my <a href="http://programmingpraxis.com>blog</a> that
> you may find useful.
>
> On Tue, Dec 11, 2012 at 10:11 AM, Stephen Bloch <sbloch at adelphi.edu>wrote:
>
>>
>> On Dec 11, 2012, at 9:03 AM, Jens Axel Søgaard wrote:
>>
>> > 2012/12/11 Stephen Bloch <bloch at adelphi.edu>:
>> >
>> >> Would it perhaps make more sense for small-primes to contain primes
>> >> themselves, in increasing order so one can be found by binary search,
>> rather
>> >> than booleans?  The O(1) behavior would be replaced by O(log(limit)),
>> but
>> >> perhaps you would save enough memory to put the limit higher.
>> >
>> > I think there are too many primes.
>> >
>> > Since
>> >
>> >> (require math)
>> >> (nth-prime 78498)
>> >    1000003
>> >
>> > there are 78497 primes below a million. On a 64 bit machine
>> > that requires  8*78497 = 627976 bytes.
>>
>> If you treated it as a vector whose elements were (compile-time-typed)
>> 32-bit ints, that would cut it by a factor of 2, but it would still be a
>> third of a megabyte.  OTOH, a vector of bit-packed booleans would take an
>> eighth of a megabyte for the same limit, and give you faster lookups.  So
>> you're right; my suggestion is probably not a win.
>>
>> How many primes are below ten million?  A hundred million?  At some point
>> storing the primes will take less memory than storing primality flags, but
>> that point may be above the size of tables we can realistically store today.
>>
>> Wait: it's conceivable that there is no such crossing point.  As the
>> numbers get big, it takes O(log(limit)) bits to store numbers less than
>> limit.  The number of primes less than limit is Theta(limit/log(limit)), so
>> storing them all takes Theta(limit) space, asymptotically the same as
>> storing the flags.
>>
>>
>>
>>
>> Stephen Bloch
>> sbloch at adelphi.edu
>>
>>
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