Question

The primes 3, 7, 109, and 673, are quite remarkable. By taking any two primes and concatenating them in any order the result will always be prime. For example, taking 7 and 109, both 7109 and 1097 are prime. The sum of these four primes, 792, represents the lowest sum for a set of four primes with this property.

Find the lowest sum for a set of five primes for which any two primes concatenate to produce another prime.

Haskell

primes :: [Int]
primes = 2 : sieve primes [3,5..] where
    sieve (p:ps) xs = h ++ sieve ps [x | x <- t, rem x p /= 0]
                      where (h, t) = span (< p*p) xs

isPrime :: Int -> Bool
isPrime n | n < 1 = False
          | otherwise = not $ or [n `rem` x == 0 | x <- [2..floor $ sqrt $ fromIntegral n]]

filterPairs :: Int -> [Int] -> [Int]
filterPairs p = filter test where
    test x = isPrime (read $ show x ++ show p) && isPrime (read $ show p ++ show x)

candidates :: [[Int]]
candidates = [[a, b, c, d, e] | a <- primes',
                                let bs = filterPairs a $ dropWhile (<= a) primes',
                                b <- bs,
                                let cs = filterPairs b $ dropWhile (<= b) bs,
                                c <- cs,
                                let ds = filterPairs c $ dropWhile (<= c) cs,
                                d <- ds,
                                let es = filterPairs d $ dropWhile (<= d) ds,
                                e <- es]
    where primes' = takeWhile (< 10000) primes

main :: IO ()
main = print $ sum $ head candidates

$ ghc -O2 -o prime-pairs prime-pairs.hs
$ time ./prime-pairs
real   0m0.351s
user   0m0.351s
sys    0m0.000s