-- https://stackoverflow.com/questions/49312746/how-to-properly-define-an-haskell-function-isprime
f n = map (mod n) [1..(round . sqrt . fromInteger) n]
primes = map fst $ filter (all (/=0).tail.snd) $ map (id &&& f) [2..]
f n = map (mod n) [2..(round . sqrt . fromInteger) n]
primes = filter (all (/=0).f) [2..]
g n = and [mod n i > 0 | i <- [2..(round . sqrt . fromInteger) n]]
primes = [ n | n <- [2..], g n]
primes = [ n | n <- [2..] , and [mod n i > 0 | i <- takeWhile ((<=n).(^2)) [2..]]]
primes = 2 : [ n | n <- [3..] , and [mod n i > 0 | i <- takeWhile ((<=n).(^2)) primes]]
primes = ps
where
ps = 2 : [n | (r:q:_, px) <- (zip . tails . (2:) . map (^2)) ps (inits ps),
n <- [r+1..q-1], all ((> 0) . rem n) px]
primes = 2 : ps
where
ps = 3 : [n | (r:q:_, px) <- (zip . tails . (3:) . map (^2)) ps (inits ps),
n <- [r+2,r+4..q-2], all ((> 0) . rem n) px]
ps = 2 : [n | (r:q:_, px) <- (zip . tails . (2:) . map (^2) <*> inits) ps,
(n,True) <- assocs ( accumArray (const id) True (r+1,q-1)
[(m,False) | p <- px, s <- [ (r+p)`div`p*p ],
m <- [s,s+p..q-1]] :: UArray Int Bool )]
For positive x, modulo x == mod x. Then,
mapMod = (\x -> map (mod x))
= (\x -> (map . mod) x)
= (map . mod)
and it is so short and clear that many would prefer just to use it right away, whether as a lambda expression or the last one, without the argument, which is known as a pointfree code ("without points" i.e. explicit arguments).
Having to remember a whole separate function name for something this short and trivial can be distracting. Thus we get
f n = map (mod n) [1..(round . sqrt . fromInteger) n]
= [mod n i | i <- [1..(round . sqrt . fromInteger) n]]
= 0 : [mod n i | i <- [2..(round . sqrt . fromInteger) n]]
= 0 : g n
g n = [mod n i | i <- [2..(round . sqrt . fromInteger) n]]
primes = filter (\n -> all (> 0) (tail $ f n)) [2..]
= [n | n <- [2..], all (> 0) (tail $ f n)]
= [n | n <- [2..], all (> 0) (tail $ 0 : g n)]
= [n | n <- [2..], all (> 0) (g n)]
= [n | n <- [2..], all (> 0)
[mod n i | i <- [2..(round . sqrt . fromInteger) n]]]
= [n | n <- [2..], all (> 0)
[mod n i | i <- takeWhile (\i-> i^2 <= n) [2..]]]
= 2 : [n | n <- [3..], all (> 0)
[mod n i | i <- takeWhile (\i-> i^2 <= n) [2..]]]
= 2 : [n | n <- [3..], all (> 0)
[mod n i | i <- takeWhile (\i-> i^2 <= n) primes]]
= 2 : [n | n <- [3..], isPrime n]
Thus we've derived the definitions
primes :: [Int]
primes = 2 : filter isPrime [3..]
isPrime n = all (> 0) [mod n i | i <- takeWhile (\i-> i^2 <= n) primes]
= and [mod n i > 0 | i <- takeWhile (\i-> i^2 <= n) primes]
= foldr (\i r -> i*i > n || (mod n i > 0 && r)) True primes
Trying it out,
~> filter isPrime [150..200] [151,157,163,167,173,179,181,191,193,197,199] ~> :sprint primes -- 13*13 = 169 primes = 2 : 3 : 5 : 7 : 11 : 13 : 17 : _ -- 17*17 = 289
primes = 2 : (`filter` [3..]) (isPrime primes)
= fix $ (2 :) . (`filter` [3..]) . isPrime
isPrime primes n = foldr (\i r -> i*i > n || (mod n i > 0 && r)) True primes