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− | == [http://projecteuler.net/index.php?section=problems&id=51 Problem 51] ==
| + | Do them on your own! |
− | Find the smallest prime which, by changing the same part of the number, can form eight different primes.
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− | Solution:
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− | millerRabinPrimality on the [[Prime_numbers]] page
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− | <haskell>
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− | isPrime x
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− | |x==3=True
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− | |otherwise=millerRabinPrimality x 2
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− | ch='1'
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− | numChar n= sum [1|x<-show(n),x==ch]
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− | replace d c|c==ch=d
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− | |otherwise=c
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− | nextN repl n= (+0)$read $map repl $show n
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− | same n= [if isPrime$nextN (replace a) n then 1 else 0|a<-['1'..'9']]
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− | problem_51=head [n|
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− | n<-[100003,100005..999999],
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− | numChar n==3,
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− | (sum $same n)==8
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− | ]
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− | </haskell>
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− | | |
− | == [http://projecteuler.net/index.php?section=problems&id=52 Problem 52] ==
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− | Find the smallest positive integer, x, such that 2x, 3x, 4x, 5x, and 6x, contain the same digits in some order.
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− | Solution:
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− | <haskell>
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− | import List
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− |
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− | has_same_digits a b = (show a) \\ (show b) == []
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− |
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− | check n = all (has_same_digits n) (map (n*) [2..6])
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− |
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− | problem_52 = head $ filter check [1..]
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− | </haskell>
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− | | |
− | == [http://projecteuler.net/index.php?section=problems&id=53 Problem 53] ==
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− | How many values of C(n,r), for 1 ≤ n ≤ 100, exceed one-million?
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− | Solution:
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− | <haskell>
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− | facs = reverse $ foldl (\y x->(head y) * x : y) [1] [1..100]
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− | comb (r,n) = facs!!n `div` (facs!!r * facs!!(n-r))
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− | perms = concat $ map (\x -> [(n,x) | n<-[1..x]]) [1..100]
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− | problem_53 = length $ filter (>1000000) $ map comb $ perms
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− | </haskell>
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− | | |
− | == [http://projecteuler.net/index.php?section=problems&id=54 Problem 54] ==
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− | How many hands did player one win in the [http://www.pokerroom.com poker games]?
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− | Solution:
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− | probably not the most straight forward way to do it.
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− | <haskell>
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− | import Data.List
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− | import Data.Maybe
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− | import Control.Monad
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− | | |
− | readCard [r,s] = (parseRank r, parseSuit s)
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− | where parseSuit = translate "SHDC"
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− | parseRank = translate "23456789TJQKA"
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− | translate from x = fromJust $ findIndex (==x) from
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− | | |
− | solveHand hand = (handRank,tiebreak)
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− | where
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− | handRank
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− | | flush && straight = 9
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− | | hasKinds 4 = 8
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− | | all hasKinds [2,3] = 7
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− | | flush = 6
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− | | straight = 5
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− | | hasKinds 3 = 4
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− | | 1 < length (kind 2) = 3
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− | | hasKinds 2 = 2
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− | | otherwise = 1
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− | tiebreak = kind =<< [4,3,2,1]
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− | hasKinds = not . null . kind
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− | kind n = map head $ filter ((n==).length) $ group ranks
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− | ranks = reverse $ sort $ map fst hand
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− | flush = 1 == length (nub (map snd hand))
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− | straight = length (kind 1) == 5 && 4 == head ranks - last ranks
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− |
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− | gameLineToHands = splitAt 5 . map readCard . words
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− | p1won (a,b) = solveHand a > solveHand b
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− | | |
− | problem_54 = do
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− | f <- readFile "poker.txt"
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− | let games = map gameLineToHands $ lines f
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− | wins = filter p1won games
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− | print $ length wins
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− | </haskell>
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− | | |
− | == [http://projecteuler.net/index.php?section=problems&id=55 Problem 55] ==
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− | How many Lychrel numbers are there below ten-thousand?
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− | Solution:
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− | <haskell>
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− | reverseNum = read . reverse . show
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− |
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− | palindrome x =
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− | sx == reverse sx
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− | where
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− | sx = show x
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− | | |
− | lychrel =
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− | not . any palindrome . take 50 . tail . iterate next
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− | where
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− | next x = x + reverseNum x
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− |
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− | problem_55 = length $ filter lychrel [1..10000]
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− | </haskell>
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− | | |
− | == [http://projecteuler.net/index.php?section=problems&id=56 Problem 56] ==
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− | Considering natural numbers of the form, a<sup>b</sup>, finding the maximum digital sum.
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− | Solution:
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− | <haskell>
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− | digitalSum 0 = 0
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− | digitalSum n =
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− | let (d,m) = quotRem n 10 in m + digitalSum d
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− |
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− | problem_56 =
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− | maximum [digitalSum (a^b) | a <- [99], b <- [90..99]]
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− | </haskell>
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− | | |
− | == [http://projecteuler.net/index.php?section=problems&id=57 Problem 57] ==
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− | Investigate the expansion of the continued fraction for the square root of two.
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− | Solution:
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− | <haskell>
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− | twoex = zip ns ds
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− | where
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− | ns = 3 : zipWith (\x y -> x + 2 * y) ns ds
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− | ds = 2 : zipWith (+) ns ds
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− |
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− | len = length . show
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− |
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− | problem_57 =
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− | length $ filter (\(n,d) -> len n > len d) $ take 1000 twoex
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− | </haskell>
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− | | |
− | == [http://projecteuler.net/index.php?section=problems&id=58 Problem 58] ==
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− | Investigate the number of primes that lie on the diagonals of the spiral grid.
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− | Solution:
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− | <haskell>
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− | isPrime x
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− | |x==3=True
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− | |otherwise=all id [millerRabinPrimality x n|n<-[2,3]]
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− | diag = 1:3:5:7:zipWith (+) diag [8,10..]
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− | problem_58 =
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− | result $ dropWhile tooBig $ drop 2 $ scanl primeRatio (0,0) diag
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− | where
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− | primeRatio (n,d) num = (if d `mod` 4 /= 0 && isPrime num then n+1 else n,d+1)
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− | tooBig (n,d) = n*10 >= d
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− | result ((_,d):_) = (d+2) `div` 4 * 2 + 1
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− | </haskell>
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− | | |
− | == [http://projecteuler.net/index.php?section=problems&id=59 Problem 59] ==
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− | Using a brute force attack, can you decrypt the cipher using XOR encryption?
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− | | |
− | Solution:
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− | <haskell>
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− | import Data.Bits
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− | import Data.Char
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− | import Data.List
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− |
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− | keys = [ [a,b,c] | a <- [97..122], b <- [97..122], c <- [97..122] ]
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− | allAlpha a = all (\k -> let a = ord k in (a >= 32 && a <= 122)) a
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− | howManySpaces x = length (elemIndices ' ' x)
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− | compareBy f x y = compare (f x) (f y)
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− |
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− | problem_59 = do
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− | s <- readFile "cipher1.txt"
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− | let
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− | cipher = (read ("[" ++ s ++ "]") :: [Int])
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− | decrypts = [ (map chr (zipWith xor (cycle key) cipher), map chr key) | key <- keys ]
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− | alphaDecrypts = filter (\(x,y) -> allAlpha x) decrypts
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− | message = maximumBy (\(x,y) (x',y') -> compareBy howManySpaces x x') alphaDecrypts
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− | asciisum = sum (map ord (fst message))
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− | putStrLn (show asciisum)
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− | </haskell>
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− | | |
− | == [http://projecteuler.net/index.php?section=problems&id=60 Problem 60] ==
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− | Find a set of five primes for which any two primes concatenate to produce another prime.
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− | Solution:
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− | Breadth first search that works on infinite lists. Breaks the 60 secs rule. This program finds the solution in 185 sec on my Dell D620 Laptop.
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− | <haskell>
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− | problem_60 = print$sum $head solve
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− | isPrime x
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− | |x==3=True
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− | |otherwise=millerRabinPrimality x 2
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− |
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− | solve = do
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− | a <- primesTo10000
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− | let m = f a $ dropWhile (<= a) primesTo10000
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− | b <- m
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− | let n = f b $ dropWhile (<= b) m
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− | c <- n
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− | let o = f c $ dropWhile (<= c) n
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− | d <- o
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− | let p = f d $ dropWhile (<= d) o
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− | e <- p
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− | return [a,b,c,d,e]
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− | where
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− | f x = filter (\y -> all id[isPrime $read $shows x $show y,
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− | isPrime $read $shows y $show x])
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− | primesTo10000 = 2:filter (isPrime) [3,5..9999]
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− | </haskell>
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