[racket] Math library ready for testing

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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