finding primes

[Shlomi Fish]

On Thu, 21 Dec 2000, S.D.Mechveliani wrote:

> Hello,
> 
> On generating prime numbers, people wrote 
> 
> ...
> |     import Array
> |     primes n = [ i | i <- [2 ..n], not (primesMap ! i)] where
> | 	primesMap   = accumArray (||) False (2,n) multList
> | [..]
> 
> 
> I think that it is good for functional programming to avoid arrays, 
> as possible. 
>

Yes. So I realized. It's a pity though, because sometimes arrays are very 
convenient.
 
> 
> Shlomi Fish <shlomif@vipe.technion.ac.il>  writes
> 
> > [..]
> > primes :: Int -> [Int]
> >
> > primes how_much = sieve [2..how_much] where
> >         sieve (p:x) = 
> >             p : (if p <= mybound
> >                 then sieve (remove (p*p) x)
> >                 else x) where
> >             remove what (a:as) | what > how_much = (a:as)
> >                                | a < what = a:(remove what as)
> >                                | a == what = (remove (what+step) as)
> >                                | a > what = a:(remove (what+step) as)
> >             remove what [] = []
> >             step = (if (p == 2) then p else (2*p)) 
> >         sieve [] = []
> >         mybound = ceiling(sqrt(fromIntegral how_much))
> >
> > I optimized it quite a bit, but the concept remained the same. 
> >
> > Anyway, this code can scale very well to 100000 and beyond. But it's not
> > exactly the same algorithm.
> > [..]
> 
> 
> C.Runciman <Colin.Runciman@cs.york.ac.uk>  gives a paper reference 
> about this.
> 
> Aslo may I ask what do you mean by "can scale to 100000",
> the value of a prime or its position No in the list?

I mean it can easily generate the list of primes from 2 up to 100000. Not
100000 separate primes.

> Anyway, here are my attempts:
> -----------------------------------
> 1.
>   primes1 = s [(2::Int)..] :: [Int]
>                      where  s (p:ns) = p: (s (filter (notm p) ns))
>                             notm p n = (mod n p) /= 0 
> 
>   main = putStr $ shows (primes1!!9000) "\n"
> 

This strikes me as the sieve version which I gave as the first example.
It's not very efficient because it uses modulo which is a costy operation.

> After compiling by  GHC-4.08 -O2
> it yields           93187  in  115 sec   on  Intel-586, 160 MHz.
> -----------------------------------

> 2.
> The DoCon program written in Haskell (again, no arrays) gives
> about 10 sec (for Integer values). 
> Also it finds, for example, first 5 primes after 10^9 as follows:
> 
>   take 5 $ filter isPrime [(10^9 ::Integer) ..] 
>   -->
>   [1000000007,1000000009,1000000021,1000000033,1000000087]
> 
> This takes  0.05 sec.
> But DoCon uses a particular  isPrime  test method:
>   Pomerance C., Selfridge J.L., Wagstaff S.S.:
>   The pseudoprimes to 25*10^9.
>   Math.Comput., 1980, v.36, No.151, pp.1003-1026.
>  
> After 25*10^9 it becomes again, expensive.
>

Where can I find this DoCon program?

Regards,

	Shlomi Fish
 
> 
> ------------------
> Sergey Mechveliani
> mechvel@botik.ru
> 
> 
> 
> 



----------------------------------------------------------------------
Shlomi Fish        shlomif@vipe.technion.ac.il 
Home Page:         http://t2.technion.ac.il/~shlomif/
Home E-mail:       shlomif@techie.com

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