{- Copyright (C) 2011 Dr. Alistair Ward This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see <http://www.gnu.org/licenses/>. -} {- | [@AUTHOR@] Dr. Alistair Ward [@DESCRIPTION@] Generates the constant, conceptally infinite, list of /prime-numbers/, using /Turner's Sieve/; <http://www.haskell.org/haskellwiki/Prime_numbers#Turner.27s_sieve_-_Trial_division>. -} module Factory.Math.Implementations.Primes.TurnersSieve( -- * Functions turnersSieve ) where import qualified Factory.Math.Power as Math.Power {- | * For each /prime/, the infinite list of candidates greater than its /square/, is filtered for indivisibility; <http://www.haskell.org/haskellwiki/Prime_numbers#Turner.27s_sieve_-_Trial_division>. * CAVEAT: though one can easily add a 'Data.PrimeWheel.PrimeWheel', it proved counterproductive. -} turnersSieve :: Integral prime => [prime] turnersSieve = 2 : sieve [3, 5 ..] where sieve :: Integral i => [i] -> [i] sieve [] = [] sieve (prime : candidates) = prime : sieve ( filter ( \candidate -> any ($ candidate) [ (< Math.Power.square prime), -- Unconditionally admit any candidate smaller than the square of the last prime. (/= 0) . (`rem` prime) -- Ensure indivisibility, of all subsequent candidates, by the last prime discovered. ] ) candidates )