module Domain.Math.Data.Primes
( primes, isPrime, coprime, primeFactors, factors
, testPrimes
) where
import Control.Monad
import Data.Function
import Data.List
import Ideas.Utils.TestSuite
import Test.QuickCheck
import qualified Data.Sequence as S
primes :: [Int]
primes = 1 : 2 : 3 : 5 : sieve (candidates 7)
primeFactors :: Int -> [Int]
primeFactors n
| n > 0 = rec (tail primes1000) n
| otherwise = error "primeFactors: non-positive argument"
where
rec [] a
| a < 1000000 = [a]
| otherwise = sort (rhos a)
rec list@(p:ps) a
| a == 1 = []
| m == 0 = p : rec list d
| otherwise = rec ps a
where
(d, m) = a `divMod` p
rhos a =
case pollardsRho a of
Just d -> rhos d ++ rhos (a `div` d)
Nothing -> [a]
primes1000 :: [Int]
primes1000 =
[1,2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97
,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193
,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293,307
,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397,401,409,419,421
,431,433,439,443,449,457,461,463,467,479,487,491,499,503,509,521,523,541,547
,557,563,569,571,577,587,593,599,601,607,613,617,619,631,641,643,647,653,659
,661,673,677,683,691,701,709,719,727,733,739,743,751,757,761,769,773,787,797
,809,811,821,823,827,829,839,853,857,859,863,877,881,883,887,907,911,919,929
,937,941,947,953,967,971,977,983,991,997]
pollardsRho :: Int -> Maybe Int
pollardsRho n = msum (map try [1..10])
where
try :: Int -> Maybe Int
try c = rec 2 2 1
where
rec :: Int -> Int -> Int -> Maybe Int
rec x y d
| d == 1 = rec nx ny (abs (nx-ny) `gcd` n)
| d == n = Nothing
| otherwise = Just d
where
nx = f x
ny = f (f y)
f :: Int -> Int
f x = (x*x+c) `mod` n
isPrime :: Int -> Bool
isPrime a =
case primeFactors a of
b:_ -> a == b
_ -> True
coprime :: Int -> Int -> Bool
coprime = rec `on` primeFactors
where
rec xs@(x:xr) ys@(y:yr) =
case compare x y of
LT -> rec xr ys
EQ -> False
GT -> rec xs yr
rec _ _ = True
factors :: Int -> [Int]
factors = sort . rec . primeFactors . abs
where
rec [] = [1]
rec (x:xs) = [ a*b | b <- take n (powers x), a <- rec zs ]
where
(ys, zs) = break (/= x) xs
n = 2 + length ys
sieveSlow :: [Int] -> [Int]
sieveSlow [] = []
sieveSlow (x:xs) = x : sieveSlow (filter (noDivisorOf x) xs)
sieve :: [Int] -> [Int]
sieve = rec S.empty
where
rec _ [] = []
rec q (x:xs) =
case S.viewl q of
(y:ys) S.:< qr | x == y ->
rec qr (ys `removeFrom` xs)
_ -> x : rec (q S.|> map (*x) (candidates x)) xs
removeFrom xs@(x:xr) ys@(y:yr) =
case compare x y of
LT -> removeFrom xr ys
EQ -> removeFrom xr yr
GT -> y:removeFrom xs yr
removeFrom _ _ = []
candidates :: Int -> [Int]
candidates n = dropWhile (< n)
[ 30*k+i | k <- [n `div` 30..], i <- [1,7,11,13,17,19,23,29] ]
divisorOf :: Int -> Int -> Bool
divisorOf x y = y `mod` x == 0
noDivisorOf :: Int -> Int -> Bool
noDivisorOf x y = y `mod` x /= 0
powers :: Int -> [Int]
powers a = iterate (*a) 1
primesSlow :: [Int]
primesSlow = 1 : 2 : sieveSlow [3, 5 ..]
testPrimes :: TestSuite
testPrimes = suite "primes"
[ assertTrue "first 1000 primes" (take 1000 primesSlow == take 1000 primes)
, assertTrue "isPrime" (all isPrime primes1000)
, useProperty "product of prime factors" $
forAll (choose (1, 1000000)) $ \n ->
product (primeFactors n) == n
, useProperty "primality of prime factors" $
forAll (choose (1, 1000000)) $ \n ->
all isPrime (primeFactors n)
, useProperty "factoring product of two primes" $
forAll (elements $ tail primes1000) $ \a ->
forAll (elements $ tail primes1000) $ \b ->
primeFactors (a*b) == sort [a, b]
, useProperty "factors" $
forAll (choose (1, 10000)) $ \n ->
all (`divisorOf` n) (factors n)
, useProperty "factors of product" $
forAll (choose (1, 1000)) $ \a ->
forAll (choose (1, 1000)) $ \b ->
all (`elem` factors (a*b)) [a, b]
]