```-- |
-- Module      : Data.Numbers.Primes
--
-- Maintainer  : Sebastian Fischer (sebf@informatik.uni-kiel.de)
-- Stability   : experimental
-- Portability : portable
--
-- This Haskell library provides an efficient lazy wheel sieve for
-- prime generation inspired by /Lazy wheel sieves and spirals of/
-- /primes/ by Colin Runciman
-- (<http://www.cs.york.ac.uk/ftpdir/pub/colin/jfp97lw.ps.gz>) and
-- /The Genuine Sieve of Eratosthenes/ by Melissa O'Neil
-- (<http://www.cs.hmc.edu/~oneill/papers/Sieve-JFP.pdf>).
--
module Data.Numbers.Primes (

primes, wheelSieve,

isPrime, primeFactors

) where

-- |
-- This global constant is an infinite list of prime numbers. It is
-- generated by a lazy wheel sieve and shared across the whole program
-- run. If you are concerned about the memory requirements of sharing
-- many primes you can call the function @wheelSieve@ directly.
--
primes :: Integral int => [int]
primes = wheelSieve 6

{-# SPECIALISE primes :: [Int]     #-}
{-# SPECIALISE primes :: [Integer] #-}

-- |
-- This function returns an infinite list of prime numbers by sieving
-- with a wheel that cancels the multiples of the first @n@ primes
-- where @n@ is the argument given to @wheelSieve@. Don't use too
-- large wheels. The number @6@ is a good value to pass to this
-- function. Larger wheels improve the run time at the cost of higher
-- memory requirements.
--
wheelSieve :: Integral int
=> Int    -- ^ number of primes canceled by the wheel
-> [int]  -- ^ infinite list of primes
wheelSieve k = reverse ps ++ map head (sieve p (cycle ns))
where (p:ps,ns) = wheel k

{-# SPECIALISE wheelSieve :: Int -> [Int]     #-}
{-# SPECIALISE wheelSieve :: Int -> [Integer] #-}

-- |
-- Checks whether a given number is prime.
--
-- This function uses trial division to check for divisibility with
-- all primes below the square root of the given number. It is
-- impractical for numbers with a very large smallest prime factor.
--
isPrime :: Integral int => int -> Bool
isPrime n | n > 1     = primeFactors n == [n]
| otherwise = False

{-# SPECIALISE isPrime :: Int     -> Bool #-}
{-# SPECIALISE isPrime :: Integer -> Bool #-}

-- |
-- Yields the sorted list of prime factors of the given positive
-- number.
--
-- This function uses trial division and is impractical for numbers
-- with very large prime factors.
--
primeFactors :: Integral int => int -> [int]
primeFactors n = factors n (wheelSieve 6)
where
factors 1 _                  = []
factors m (p:ps) | m < p*p   = [m]
| r == 0    = p : factors q (p:ps)
| otherwise = factors m ps
where (q,r) = quotRem m p

{-# SPECIALISE primeFactors :: Int     -> [Int]     #-}
{-# SPECIALISE primeFactors :: Integer -> [Integer] #-}

-- Auxiliary Definitions
------------------------------------------------------------------------------

-- Sieves prime candidates by computing composites from the result of
-- a recursive call with identical arguments. We could use sharing
-- instead of a recursive call with identical arguments but that would
-- lead to much higher memory requirements. The results of the
-- different calls are consumed at different speeds and we want to
-- avoid multiple far apart pointers into the result list to avoid
-- retaining everything in between.
--
-- Each list in the result starts with a prime. To obtain composites
-- that need to be cancelled, one can multiply all elements of the
--
sieve :: (Ord int, Num int) => int -> [int] -> [[int]]
sieve p ns@(m:ms) = spin p ns : sieveComps (p+m) ms (composites p ns)

{-# SPECIALISE sieve :: Int     -> [Int]     -> [[Int]]     #-}
{-# SPECIALISE sieve :: Integer -> [Integer] -> [[Integer]] #-}

-- Composites are stored in increasing order in a priority queue. The
-- queue has an associated feeder which is used to avoid filling it
-- with entries that will only be used again much later.
--
type Composites int = (Queue int,[[int]])

-- The feeder is computed from the result of a call to 'sieve'.
--
composites :: (Ord int, Num int) => int -> [int] -> Composites int
composites p ns = (Empty, map comps (spin p ns : sieve p ns))
where comps xs@(x:_) = map (x*) xs

{-# SPECIALISE composites :: Int     -> [Int]     -> Composites Int     #-}
{-# SPECIALISE composites :: Integer -> [Integer] -> Composites Integer #-}

-- We can split all composites into the next and remaining
-- composites. We use the feeder when appropriate and discard equal
-- entries to not return a composite twice.
--
splitComposites :: Ord int => Composites int -> (int,Composites int)
splitComposites (Empty, xs:xss) = splitComposites (Fork xs [], xss)
splitComposites (queue, xss@((x:xs):yss))
| x < z     = (x, discard x (enqueue xs queue, yss))
| otherwise = (z, discard z (enqueue zs queue', xss))
where (z:zs,queue') = dequeue queue

{-# SPECIALISE splitComposites :: Composites Int -> (Int,Composites Int) #-}
{-# SPECIALISE
splitComposites :: Composites Integer -> (Integer,Composites Integer) #-}

-- Drops all occurrences of the given element.
--
discard :: Ord int => int -> Composites int -> Composites int
| otherwise = ns
where (m,ms) = splitComposites ns

{-# SPECIALISE discard :: Int -> Composites Int -> Composites Int #-}
{-# SPECIALISE
discard :: Integer -> Composites Integer -> Composites Integer #-}

-- This is the actual sieve. It discards candidates that are
-- composites and yields lists which start with a prime and contain
-- all factors of the composites that need to be dropped.
--
sieveComps :: (Ord int, Num int) => int -> [int] -> Composites int -> [[int]]
sieveComps cand ns@(m:ms) xs
| cand == comp = sieveComps (cand+m) ms ys
| cand <  comp = spin cand ns : sieveComps (cand+m) ms xs
| otherwise    = sieveComps cand ns ys
where (comp,ys) = splitComposites xs

{-# SPECIALISE sieveComps :: Int -> [Int] -> Composites Int -> [[Int]] #-}
{-# SPECIALISE
sieveComps :: Integer -> [Integer] -> Composites Integer -> [[Integer]] #-}

-- This function computes factors of composites of primes by spinning
-- a wheel.
--
spin :: Num int => int -> [int] -> [int]
spin x (y:ys) = x : spin (x+y) ys

{-# SPECIALISE spin :: Int     -> [Int]     -> [Int]     #-}
{-# SPECIALISE spin :: Integer -> [Integer] -> [Integer] #-}

-- A wheel consists of a list of primes whose multiples are canceled
-- and the actual wheel that is rolled for canceling.
--
type Wheel int = ([int],[int])

-- Computes a wheel that cancels the multiples of the given number
-- (plus 1) of primes.
--
-- For example:
--
-- wheel 0 = ([2],[1])
-- wheel 1 = ([3,2],[2])
-- wheel 2 = ([5,3,2],[2,4])
-- wheel 3 = ([7,5,3,2],[4,2,4,2,4,6,2,6])
--
wheel :: Integral int => Int -> Wheel int
wheel n = iterate next ([2],[1]) !! n

{-# SPECIALISE wheel :: Int -> Wheel Int     #-}
{-# SPECIALISE wheel :: Int -> Wheel Integer #-}

next :: Integral int => Wheel int -> Wheel int
next (ps@(p:_),xs) = (py:ps,cancel (product ps) p py ys)
where (y:ys) = cycle xs
py = p + y

{-# SPECIALISE next :: Wheel Int     -> Wheel Int     #-}
{-# SPECIALISE next :: Wheel Integer -> Wheel Integer #-}

cancel :: Integral int => int -> int -> int -> [int] -> [int]
cancel 0 _ _ _ = []
cancel m p n (x:ys@(y:zs))
| nx `mod` p > 0 = x : cancel (m-x) p nx ys
| otherwise      = cancel m p n (x+y:zs)
where nx = n + x

{-# SPECIALISE cancel :: Int -> Int -> Int -> [Int] -> [Int] #-}
{-# SPECIALISE
cancel :: Integer -> Integer -> Integer -> [Integer] -> [Integer] #-}

-- We use a special version of priority queues implemented as /pairing/
-- /heaps/ (see /Purely Functional Data Structures/ by Chris Okasaki).
--
-- The queue stores non-empty lists of composites; the first element
-- is used as priority.
--
data Queue int = Empty | Fork [int] [Queue int]

enqueue :: Ord int => [int] -> Queue int -> Queue int
enqueue ns = merge (Fork ns [])

{-# SPECIALISE enqueue :: [Int]     -> Queue Int     -> Queue Int     #-}
{-# SPECIALISE enqueue :: [Integer] -> Queue Integer -> Queue Integer #-}

merge :: Ord int => Queue int -> Queue int -> Queue int
merge Empty y                        = y
merge x     Empty                    = x
merge x     y     | prio x <= prio y = join x y
| otherwise        = join y x
where prio (Fork (n:_) _) = n
join (Fork ns qs) q = Fork ns (q:qs)

{-# SPECIALISE merge :: Queue Int     -> Queue Int     -> Queue Int     #-}
{-# SPECIALISE merge :: Queue Integer -> Queue Integer -> Queue Integer #-}

dequeue :: Ord int => Queue int -> ([int], Queue int)
dequeue (Fork ns qs) = (ns,mergeAll qs)

{-# SPECIALISE dequeue :: Queue Int     -> ([Int],     Queue Int)     #-}
{-# SPECIALISE dequeue :: Queue Integer -> ([Integer], Queue Integer) #-}

mergeAll :: Ord int => [Queue int] -> Queue int
mergeAll []       = Empty
mergeAll [x]      = x
mergeAll (x:y:qs) = merge (merge x y) (mergeAll qs)

{-# SPECIALISE mergeAll :: [Queue Int]     -> Queue Int     #-}
{-# SPECIALISE mergeAll :: [Queue Integer] -> Queue Integer #-}
```